Where Fermi Fails: What is hard to estimate?

The rate of change in atmospheric pressure in Moscow exactly 1 year from now in mbar/hour.

Presumably deterministic chaos-based problems, like the famous 3-body problem would qualify, or anything relying on the butterfly effect.

Edit: this was unintentionally redacted!! Is there a undo-the-redaction option?

Sally and John both play a videogame where their characters live on a circle. Starting next to Sally, John walks around the circle 10 to the 100th times (he is a known hacker). Sally hasn't moved and measures the position of John. Assuming that Sally is at the point (1,0) and John moves counter-clockwise, what is the slope of the trajectory of the projectile that John's character fires towards the center of the circle?

For a real one: what is the length of the coast of Britain? Perhaps this is ill-defined without giving a length-scale, but even with a scale (say, a meter) I'm not sure it's doable. Or similarly, what's the surface area of a convenient piece of broccoli?

Busy beaver is a little wordy to state in 10 seconds, but is nonetheless a good example.

(And my example in OP has been revised in light of this oversight.)

Mathematica says the exact value of y that maximizes the distance between arctan(1.1y) and arctan(0.9y) is 1/sqrt(0.99) which is approximately 1.00504 (its negative also works). Feynman should still choose 1, because this isn't a calculation to do in your head, and 1 is a good guess.

His chances of winning, however, are about 3.1%: the range of x-values that work is roughly 0.1, and the period of tangent is pi.

I got the correct GCD, but then realized that by writing out the steps in Notepad I violated the no-computer injunction :(

Without looking it up (or clicking that link): 54,000,000?

(Vaguely remembering that it's about n log n (maybe?), and attempting to approximate log in my head.)

(EDIT: I was almost within 20%... but even that approximation above isn't within 10%.)

IIRC, density(areogel) - density(air) < density(air) - density(helium), so I guess it would.

Maybe this is a trick question, but why wouldn't it float?

"Busy beaver billion."

As for the Earth, I'd say light radionuclides such as carbon-14 are more or less in equilibrium (as many are created by cosmic rays as many decay), and so are heavy short-lived radionuclides such as radon (as many are created by uranium and thorium decaying as many decay themselves), whereas heavy long-lived radionuclides (uranium and thorium) decay with nothing replenishing them. So the total radioactivity is decreasing (at least on timescales long compared with the 2x11-year solar cycle but short compared with the Earth's age).

R(5, 5) is hard to solve exactly, but easy to estimate: 46 is correct to within 10%.

(Although perhaps only Feynman could produce such an estimate in ten seconds, without knowing the range beforehand.)

(For the record)

Python 3.2.2 (default, Sep  5 2011, 21:17:14)  
[GCC 4.6.1] on linux2  
Type "help", "copyright", "credits" or "license" for more information.  
>>> from math import tan, pi  
>>> tan(10**100)  
-0.4116229628832498  
>>> tan((180/pi)*(10**100)) # degrees  
2.415162133199225