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.
