TITLE: Naturally occurring orderings
QUESTION [9 upvotes]: The are many orderings that naturally occur in interesting but seemingly unrelated circumstances. Here are some examples:

The volume spectrum of orientable hyperbolic 3-manifolds has order type $\omega^\omega$.
Ordinals that play important roles in Conway's $\mathbf {On_2}$, most notably $\omega^{\omega^\omega}$, the algebraic closure of $2$. See Lenstra's papers 1 2, Conway's ONAG, and Lieven's blog posts.
The set of fusible numbers has order type $\epsilon_0$  (quite likely but not proven, see my note).
The Sharkovsky ordering of natural numbers, which does not have order type of an ordinal.
There are proof theoretical ordinals, which I know little about.

Do you know any other examples or see any connection among aforementioned examples? Most of the examples above are ordinals, but other interesting examples are welcome.

REPLY [9 votes]: Alright, I'll put my comment as an answer and hopefully get this off the no-upvoted-answers queue. :)
Here's another nice-but-surprising way to get $\omega^\omega$: Let $\|n\|$ denote the smallest number of 1's needed to write n using any combination of addition and multiplication, e.g., $\|7\|=6$ as shortest way for 7 is $7=(1+1+1)(1+1)+1$.  (This is known as the "integer complexity" of n; it's sequence A005245.)
Now, for any n, we have the lower bound $\|n\|\ge 3log_3 n$. So subtract this off and consider $\delta(n):=\|n\|-3log_3 n$.  Then the set of all values of $\delta$ is a well-ordered subset of $\mathbb{R}$, with order type $\omega^\omega$.
For a proof, I refer you to my preprint: http://arxiv.org/abs/1310.2894