Surprising places that particular countable ordinals show up. In this paper, a rather simply defined set (the smallest set $S$ such that $0 \in S$ and $x, y \in S \implies(x+y+1)/2 \in S$ whenever $|y-x| \le 1$) motivated by a riddle is found to have order type $\epsilon_0$. I found it quite impressive that a large (to me, at least) countable ordinal would show up here. Are there many other instances where countable ordinals show up in surprising contexts?
 A: So, I have some examples, although the surprise is mostly in that the set is well-ordered, rather than what the ordinal is; in fact, all of these cases will have order type either $\omega^\omega$ or conjectured to be $\omega^\omega$.

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*Volumes of hyperbolic 3-manifolds.  I don't know anything about the mathematics of this, but these are apparently well-ordered, with order type $\omega^\omega$.  The set is also closed.


*Commuting probabilities in finite groups.  Consider a finite group $G$; what's the probability that two elements drawn from it at random commute?  Sean Eberhard showed that this set is reverse-well-orderd, and its reverse order type is either $\omega^\omega$ or $\omega^{\omega^2}$.  (I asked him and he said he thinks the former is probably the correct ordinal, but this remains unproven to my knowledge.)
Of note, one could speculatively generalize this in multiple ways; e.g., what about the probability of satisfying words other than $aba^{-1}b^{-1}$?  To my knowledge no other nontrivial case remains proven.  One could also ask about compact groups rather than finite groups; in the case of commuting probabilities, Eberhard has shown that compact groups give the same probabilities as finite groups (other than zero).
This set is also conjectured to be closed, although if it is closed, it would have to be for a weird reason.


*To toot my own horn -- Let $||n||$ denote the smallest number of 1's needed to write $n$ using any combination of addition and multiplication.  It's known that $||n||\ge 3\log_3 n$.  I showed that the values of $||n||-3\log_3 n$ are well-ordered with order type $\omega^\omega$.  Also (and I don't have a reference for this result because I'm still writing it up!) the closure is the set of numbers of the form $||n||+k-3\log_3 n$ (which of course has the same order type).

But that's not all!  Because if we look at addition chains instead, letting $\ell(n)$ be the length of the smallest addition chain for $n$, then the values of $\ell(n)-\log_2 n$ are also well-ordered with order type $\omega^\omega$.  In this case I haven't proven that the closure takes the form above.
I would speculate that such well-ordering results will likely hold for a number of other variants of the problem, but I can't prove it at the moment for most of them.


*Finally let me mention a case that, despite not being well-ordered, I think is quite noteworthy anyway.  This Pisot numbers are closed and not well-ordered, but their order type is a very interesting one, and one I like to think of as being a non-well-ordered analogue of $\omega^\omega$.  It's an order type I expect can be found in more places, and I have some ideas where we can expect it to show up. :)

So, I hope these answers are sufficiently surprising, even though it's all the same ordinal! :)
Edit: Here's one more; apparently the set of dynamical degrees of birational transformations of the complex projective plane is closed and well-ordered.  However this paper doesn't seem to offer any information about its order type; this is not my area at all so if more is now known I don't know about it.  This also seems like the sort of thing that would likely be susceptible to generalization, possibly.
Further edit: I asked the authors and they seem to think this one is another $\omega^\omega$, though it doesn't seem that anyone's proven it as yet.
