Does the body have an almost infinite number of potential positions?

One of the issues with the sentence is that "potential position" seems ill-defined. Even with a single joint you can have an infinite number of potential positions because you have an infinite amount of numbers between 0 and 1.

You would need to have a standard to distinguish two positions from each other to make positions countable in a meaningful sense. If you for example care about dancing you might say that two positions are the same if they have the same labanotion. A osteopath however cares about difference in articulation of a joint that are a lot more fine then what you could write down in labanotation.

Perhaps this is what is true: However many postures and movements and ways of thinking about them and experiencing them you learn, the space of possibilities will remain unexhausted. For all practical purposes, the possibilities are unlimited: no-one will have cause to lament that there is nothing left to discover.

I don't know what "almost infinite" means, but yes, the body has an infinite number of potential positions. E.g. you could raise your arm by any angle from 0 to 180 degrees. There are infinitely many real numbers from 0 to 180, hence infinitely many possible body positions.

If you mean self-controlled conscious positioning, I would disagree. There are limits to our sense of proprioception, fine motor control, etc. So, I doubt it's possible consciously move our body into an infinite number of positions. That being said, I do think we can consciously move our body into a huge number of different positions (even of no practical use) but, the number is not infinite or approaching infinite.

Alternatively...

If you mean things beyond just joints, muscles, etc., like changes in the position of individual cells due to breathing, changes in blood flow/pressure, and things like that. I would guess that would approach an infinite number of positions. There are roughly 37.2 trillion cells in the human body, changes in their positioning would add up quick :-)

Depends on your perspective...

No, not infinite, but in a technical sense very very high, in the same way that the Coastline Paradox allows you to have lots and lots of measurements for the length of a coastline. Since you can get to arbitrary precision with your measurements (not infinitely arbitrary, but like, really really really high), you can get more and more positions. And unlike the coastal paradox, you have multiple dimensions you can change which causes the total number of positions to balloon.

However, in a practical sense, for a given application you're going to have some level of precision that you want, and some limbs/body areas that you care about, this level and focus will limit the total number of possible positions for that purpose. For instance, sitting in a chair limits the amount of movements you can make, and then in terms of precision you probably alone care about gross postural positions rather than minute details of degree, so in practice there might be only 10 or so relevant "chair posture" positions.