Each potion has two essential ingredients (necessary, but not sufficient).

Barkskin Potion: Crushed Onyx and Ground Bone

Farsight Potion: Beholder Eye and Eye of Newt

Fire Breathing Potion: Dragon Spleen and Dragon's Blood

Fire Resist Potion: Crushed Ruby and Dragon Scale

Glibness Potion: Dragon Tongue and Powdered Silver

Growth Potion: Giant's Toe and Redwood Sap

Invisibility Potion: Crushed Diamond and Ectoplasm

Necromantic Power Potion: Beech Bark and Oaken Twigs 

Rage Potion: Badger Skull and Demon Claw

Regeneration Potion: Troll Blood and Vampire Fang

Most of these make sense. Except...

I strongly suspect that Archmage Anachronos is trying to trick me into getting him to brew a Necromantic Power Potion, and has swapped around the "Barkskin Potion" and "Necromantic Power Potion" output reports. In character, I should obtain information from other sources to confirm this. For the purpose of this problem, I will both try to solve the stated problem but also how to make a "Necromantic Power Potion" (i.e. an actual Barkskin Potion) to troll Anachronos.

Barksin and Necromantic Power also seem to be the two toughest potions to make, never succeeding with just a third ingredient added.

An attempt that includes essential ingredients for multiple potions does not necessarily fail, some ingredient combinations can produce multiple potion types.

There are four failure modes:

Acidic Slurry never happens if the essential ingredients of any potion are included, but not all attempts that lack essential ingredients are Acidic Slurry. So, I'm guessing Acidic Slurry is a residual if it doesn't have the required ingredients for any potion and doesn't hit any of the other failure modes.

Inert Glop tends to happen with low numbers of ingredients, initial guess is it happens if "not magical enough" in some sense, and guessing Magical Explosion is the opposite, dunno about Mutagenic Ooze yet.

Back to Barkskin and Necromantic Power:

Either one has lots of options to make reliably with just one non-available ingredient, but not (in the stats provided) avoiding all the unavailable ingredients. Each has some available options that produce the potion type most of the time, but with Inert Glop some of the time. There's one ingredient combination that has produced a "Barkskin Potion" the only time it has been tried, but it has the essential ingredients for a Growth Potion so would likely not reliably produce a "Barkskin Potion".

Many ingredient combos especially towards higher ingredient numbers haven't been tried yet, so plenty of room to find an actually reliable solution if i can figure out more about the mechanics. The research will continue...

Looks like you only get magical explosions and mutagenic ooze with >=4 ingredients, and also like potions are determined by a specific pair of ingredients. List is the next spoiler:

Farsight: Beholder Eye and Eye of Newt
Firebreathing: Dragon Spleen and Dragon's Blood
Fire Resistance: Crushed Ruby and Dragon Scale
Glibness: Dragon Tongue and Powdered Silver
Growth: Giant Toe and Redwood Sap
Invisibility: Crushed Diamond and Ectoplasm
Necromantic Power: Beech Bark and Oaken Twigs
Rage: Badger Skull and Demon Claw
Regeneration: Troll Blood and Vampire Fang

Barkskin: Crushed Onyx and Ground Bone - both of which we have!

So... why aren't we just done? Why don't we just tell Anachronos to add Crushed Onyx and Ground Bone and have done with it?

Well: I still don't know yet what makes a brew succeed or fail.

My working hypothesis is that a major thing determining what happens when you try to brew a potion is what pairs of ingredients are found in it.

I suspect it's even deterministic - i.e., the same subset of potion ingredients results in the same outcome every time - but I haven't yet actually checked that - I'm doing all this in Google Sheets and my data-manipulation-fu is pathetically weak. (Probably someone could check this easily and I look dumb; I'll poke at looking for identical ingredient-sets resulting in different outcomes later.) Lorxus from like an hour later sez: "Nope, decidedly nondeterministic, given the filter-based checking I did later. Still gotta figure out what causes it and why."

For instance, I suspect that Inert Glop is the default outcome in some sense, i.e., you get Inert Glop IF no pair of ingredients in the ingredient-set codes for any potion AND no more interesting failure condition is met. Acidic Slurry would be the next-to-default, where nothing more interesting happened BUT something was different such that the result of the brew was AS and not IG.

I suspect this specifically because AS and IG are the two simplest results - each show up in the result table with only 3 ingredients.

Poking a little more with some better use of filters, it looks like my earlier theory was maybe incomplete. It's true that e.g. Barkskin Potion requires Crushed Onyx and Ground Bone, but that's not actually sufficient.

Sometimes you add those to a brew and you get nothing. An earlier working hypothesis I had that maybe Magical Explosions happen to a brew precisely when it codes for two or more potions got falsified - there's plenty of brews with e.g. Crushed Onyx and Ground Bone and also Dragon Spleen and Dragon's Blood. Some of those make Barkskin Potions. Some of them make Firebreathing potions. Sometimes you get a Mutagenic Ooze or a Magical Explosion.

So maybe that falsified theory was less falsified than I thought, and there's something about a failure rate (and likely also type) dependent on how many of the valid potions your brew codes for? Maybe something like "if a brew codes for at least two valid potions, then there's a 25% chance of a ME, a 25% chance of a MO, and equal chances of every other valid potion your brew codes for."

I didn't note this above but notably, four ingredients don't code for any of the potions: Angel Feather, Crushed Sapphire, Faerie Tears, and Quicksilver. I can't legibly explain why, but I feel like they'll still matter - have an effect on failure types or rates, or be required as part of a potion base, or maybe even just very clearly have no effect at all.

As a terminological note you may or may not have picked up on by now: I'm going to use "brew" for some subset of potion ingredients, "potion" for one of the 10 potions we can brew, and say that a brew "codes for a potion" if a subset of the brew is a pair that looks to me to determine which of the 10 potions get brewed. The "outcomes of a brew" is the set of things that have observably resulted from a given brew.

Did a quick observational-experiment:

Among brews consisting of Crushed Onyx, Dragon Spleen, Dragon's Blood, Ground Bone, and Powdered Silver, the outcomes consisted of 4 Barkskin, 9 Firebreathing, 20 Inert Glop, 8 Mutagenic Ooze, and 0 Magical Explosion.

Among brews consisting of Beech Bark, Beholder Eye, Eye of Newt, Troll Blood, and Vampire Fang, the outcomes consisted of 3 Farsight, 4 Regeneration, 0 Inert Glop, 11 Mutagenic Ooze, and 0 Magical Explosion.

(this is the point at which if my data-manipulation-fu were stronger, I'd probably just generate all possible results for all brews of the form [Putative Coding-Pair for Potion 1] + [Putative Coding-Pair for Potion 2] + [Arbitrary Other Ingredient] and look at the spread of results; I would strongly expect that to be something like "sometimes you get potion 1, sometimes you get potion 2, and there's patterns in the ratios between them and the failure rates; the failure rates, summed up, are always greater than the chance of getting any individual potion and are generally greater than getting any successful potion at all".)

...actually, that means that I (think that I) understand this problem well enough to make an initial solution-proposal! 

OK OK less of a solution-proposal, more of what has turned into another observational-experiment. What happens if we filter by requiring Crushed Onyx and Ground Bone (like we have to) and forbidding the ingredients we don't have any of anyway?^[1] Basically, let's just look at the observational-space of brews that already seem kind of like a good idea anyway - what happens in all of those possible-worlds?

We get 201 brews, and we should already spot poor Anachronos's problem: none of the reliably working 3-ingredient brews that make Barkskin are found among them.

Perhaps worse, of the 4-ingredient brews, 19 make Barkskin and 44 fail as Inert Glop; a lot of these are the specific brew {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}, which is responsible for all of the successes and 10 of the failures for unclear reasons. He was right to call through the void at right angles to try to find Math-Doers, damn the risks!

Another promising-looking brew among the observed ones is the 5-ingredient brew {[Crushed Onyx, Ground Bone], Giant's Toe, (Troll Blood, Vampire Fang)}, which makes Barkskin 18 times, Regeneration 22 times, and Mutagenic Ooze 35 times. Maybe Anachronos likes those odds and a Regen potion will do in a pinch?

So I guess I'd say if I were under truly tight time constraints I'd present Anachronos with those two options - [a ~2/3 chance of straight out success and a ~1/3 chance of harmless failure] vs [a ~1/4 chance of straight out sucess, a ~1/4 chance of lesser failure, and a ~1/2 chance of dangerous failure] and probably push him towards the former of the two - {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}. I have the luxury of existing outside of his time, though, so I'm going to keep trying.

Interestingly - the modified brew {[Crushed Onyx, Ground Bone], Demon Claw, Troll Blood} only gets tested once - a failure. That's kind of a shame - I'd predict it to be pretty good given the lack of potion-code collision with Regen (which the 5-ingredient brew suffers from). That means there's something more at work here. {[Crushed Onyx, Ground Bone], Giant's Toe, Vampire Fang} never even gets tested at all!

Note to future Lorxus - run this same process with the three other potions that we could possibly brew at all - Growth, Rage, and Regen.

Growth:
{[Giant's Toe, Redwood Sap], Quicksilver, Vampire Fang}: 24 Growth, 29 IG
{[Giant's Toe, Redwood Sap], Ground Bone, Oaken Twig, Quicksilver, Vampire Fang}: 19 Growth, 16 IG

Rage:
{[Badger Skull, Demon Claw], Quicksilver, Troll Blood}: 21 Rage, 13 IG
(notably, modified brews with only Quicksilver or Troll Blood fail more often, and fail altogether if they have nothing else - this looks to be a pattern that holds up of secondary ingredients? "Require A and B; require at least one of C and D plus something else - maybe one of the four miscellaneous ones?")

Regen:
{[Troll Blood, Vampire Fang], Crushed Diamond, Giant's Toe, Oaken Twig}
{[Troll Blood, Vampire Fang], Crushed Diamond, Demon Claw, Ground Bone}
(neither of these brews ever fail???)
 

Ah, the joys of existing outside of (Anachronos's) space-time!

I think I've squeezed about as much as I can from pure associational observation, and need to actually buckle down and figure out why brews succeed or fail. Here's my working model:

• There are precisely ten codeable potions: Barkskin, Farsight, Firebreathing, Fire Resistance, Glibness, Growth, Invisibility, Necromantic Power, Rage, and Regeneration.
  • For each of these potions, there's two ingredients that are absolutely required (the potion's code, in my terms). I list them off above.
• If a brew codes for two potions, it has a 1/4 chance of producing potion A, a 1/4 chance of producing potion B, and a 1/2 chance of melting down into Mutagenic Ooze.
  • I also hypothesize one of three things:
    • If a brew codes for n>2 potions, it has a 1/2n chance of producing each of those potions, and a 1/2 chance of melting down into Mutagenic Ooze.
    • If a brew codes for n>2 potions, it has a 1/n+2 chance of producing each of those potions, and a 2/n+2 chance of melting down into Mutagenic Ooze.
    • If a brew codes for three or more potions, then whenever it would turn out as Mutagenic Ooze, it instead turns out as a Magical Explosion. (This one I think is much less likely, given that none of the observed brews that we could replicate result in a ME.)
• A successful brew cannot, for whatever reason, consist of any fewer than 3 ingredients.
  • 4-5 in particular seems to be a sweet spot: 2 ingredients for the code, and another 2-3 for ??reasons?? which also don't code for any different potion.
• No brew is guaranteed to succeed. Every potion has some base chance of failing as Inert Glop.
  • As a corollary, we should thus reject any brew with a success rate less than that of {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang}, that is, ~2/3.
  • This is likely wrong though! {[Troll Blood, Vampire Fang], Crushed Diamond, Giant's Toe, Oaken Twig} and {[Troll Blood, Vampire Fang], Crushed Diamond, Demon Claw, Ground Bone} both seem to be no-fail brew for Regen...
• A brew which codes for no potion... gods I wish my data-manipulation-fu were stronger
  • ...is guaranteed to be Acidic Slurry? Maybe?
    • No, that can't be right - {Ground Bone, Oaken Twigs, Quicksilver, Vampire Fang} doesn't code for anything and yet we see it in the IG list.
  • ...will turn out as IG if it lacks some factor and as AS if it has that factor?
    • Yeah this isn't a guess so much as an entire guessing schema.
      • Which still might not even be correct.

So... maybe I want to go looking for relations in or commonalities among those other 2-3 ingredients? Or possibly there's specific peculiar properties to each ingredient that I need to keep track of? ...Actually that sounds a lot more plausible when I think about it that way.

Y'know... there's enough test cases that the proportions I found in the last recipe should really be a lot closer to exact than they are. What's up with that?

Also, I still have no idea why MEs happen. What's up with those?

A quick look at the set of observed brews we could go make right now that resulted in MEs:
{Crushed Onyx, Demon Claw, Giant's Toe, Oaken Twig, Redwood Sap, Troll Blood, Vampire Fang}: 16 MEs, 13 MOs, 4 Growth, 9 Regen
{Demon Claw, Giant's Toe, Troll Blood, Vampire Fang}: 1 ME/1 brew (!)

So maybe there's something special about these four ingredients? You (can) get a ME if you include all four, with the rate of that increasing as the proportion of "magically charged" ingredients in a brew increases?

It'd explain why we probably can't just tell Anachronos to brew {Crushed Onyx, Ground Bone}: it wouldn't have sufficient "magical charge"; in this model, a potion goes off (in the sense of "is not IG") never/sometimes/always if it has 0-1/2/3+ magically charged ingredients.

The problem is that I'll need to expand my search to the rest of the dataset (of things we can't possibly brew now) to figure out whether Crushed Onyx is magically charged, given that it's not in the codes for Growth or Regen (because codes are nonoverlapping) - though if that theory is correct, the existence of sometimes-working 3-ingredient brews that can make Barkskin or IG suggests that Crushed Onyx probably is magically charged (and if so, good gods, Anachronos, what were you doing brewing the same maximally overcharged potion that many times?). Also, countervailing that is the fact that {[Crushed Onyx, Ground Bone], Demon Claw, Vampire Fang} doesn't always actually work, which suggests that it's only the smaller set of four (out of what we have left).

Assuming Crushed Onyx is magically charged: {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw}.
Assuming Crushed Onyx is not magically charged: {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw, Troll Blood} and {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw, Vampire Fang} should both work equally well - all we need to do is avoid code-clash with Regen.

If I had to give a solution right now - given the fact that Troll Blood and Vampire Fang are absent from all of the (4) {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw}-containing brews, and those tend to fail as IG (or MO from code-clash) - I'm pretty sure Crushed Onyx is magically uncharged.

So I'd tell Anachronos to brew {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw, Troll Blood} or {[Crushed Onyx, Ground Bone], Giant's Toe, Demon Claw, Vampire Fang}, whichever [he feels better about]/[is cheaper]/[he likes better]/[he has better stocks of]/[he can buy more easily]/[idk my dude flip a coin]. Either should work with certainty.

https://docs.google.com/spreadsheets/d/1kzR25lXiGqhSMg6K_QwDfoE0VnVUnOeK74AbzOSkjmk/edit?usp=sharing

(after reading simon's observation that maybe Barkskin and Necromantic Power are swapped around) If we do in fact doubt Anachronos here, then we can easily put together a 5-ingredient brew that would guarantee getting him actual Barkskin: {[Beech Bark, Oaken Twig], Giant's Toe, Demon Claw, [Troll Blood XOR Vampire Fang]}

Crushed Onyx, Ground Bone, Demon Claw, Giant's Toe, Vampire Fang

First two ingredients specify the potion, remaining three make it juuust impossible enough to guarantee that it will reliably be magical without going boom.

With the ingredients he has, he has gotten a successful Barkskin Potion:

1 of the 1 times (100%) he brewed together Crushed Onyx, Giant's Toe, Ground Bone, Oaken Twigs, Redwood Sap, and Vampire Fang. 

19 of the 29 times (66%) he brewed together Crushed Onyx, Demon Claw, Ground Bone, and Vampire Fang.

Only 2 other combinations of the in-stock ingredients have ever produced Barkskin Potion, both at under a 50% rate (4/10 and 18/75).

The 4-ingredient, 66% success rate potion looks like the best option if we're just going to copy something that has worked. That's what I'd recommend if I had to make the decision right now.

Many combinations that used currently-missing ingredients reliably (100%) produced Barkskin Potion many times (up to 118/118). There may be a variant on one of those, which he has never tried, that could work better than 66% of the time using ingredients that he has. Or there may be information in there about the reliability of the 6-ingredient combination which worked once.

All potions have between 3 and 8 ingredients. Those with 6 and 7 are most liekly to succeed with 3 being particularly unlikely. In particular Barkskin Potion is never brewed successfully with only 3.
Barkskin Potion, in common with all other potions has two ingredients that must be present, in this case Crushed Onyx and Ground Bone.
 
The same set of ingredients sometimes produce different potions so there is clearly some randomness involved.
 
Scanning through the results there are several combinations that always produce the required potion. The most convincing (With 118 entries) is:
Crushed Onyx
Dragon Tongue
Dragons Blood
Eye of Newt
Giants Toe
Ground Bone
 
Sadly 3 of these ingredients aren't available however. The same is true of most of the combinationsn that have a 100 percent success rate. There is only one exception but as that is based on a single record using that is highly risky.
 
The next best with a 65.5 percent chance of success based on 29 records is:
Crushed Onyx
Demon Claw
Ground Bone
Vampire Fang 
 
So I'm going to have to recommend he uses these, and hope he doesn't end up with Inert Glop instead. 

From looking at every ingredient-result pair and picking the ones which appear with Barkskin Potion more often than would expected if they were completely independent, I'm going to suggest the ingredients
1. Crushed Onyx
2. Demon Claw
3. Ground Bone
4. Quicksilver
5. Troll Blood
I'm very certain about Crushed Onyx and Ground Bone which appear in every successful Barkskin Potion, less so about the rest.

Edit: if I filter to only successful Barkskin Potions without any of the unavailable ingredients, that leaves 42 rows. None of them use Quicksilver, and instead Giant's Toe and Vampire Fang are looking pretty good. Not sure how much weight to put on this but something to investigate later.