Naturally occurring orderings 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.
 A: 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
A: Hrbacek, following Ballard, has recently put a certain partial ordering called $\sqsubseteq$ on absolutely everything in order to do nonstandard analysis a la Nelson (i.e. internal set theory): 
Let $\mathscr{L}$ be the language of ZFC (and Tarski-Grothendieck if you insist). We say a well-formed formula of $\mathscr{L}$ is an $\in$-formula. We assert that ZFC holds for all well-formed $\in$-formulae.
Now we throw in $\sqsubseteq$ to $\mathscr{L}$ to get a bigger language $\mathscr{HB}.$ First, we assume that ZFC holds for all $\in$-formulae. Let us write 
$x \sqsubseteq_\alpha y$ as an abbreviation of 
$x \sqsubseteq \alpha \vee x \sqsubseteq y.$ 
Suppose we have a well-formed formula $P$ of $\mathscr{HB}.$ We write $P^\alpha$ for the replacement of every instance of $\sqsubseteq$ with $\sqsubseteq_\alpha.$
Let us also write $x \sqsubset y$ for
$x \sqsubseteq y \wedge y \not\sqsubseteq x.$
We also write $2^A_{\mathrm{fin}}$ for the set of all finite subsets of $A.$
We also write $(\forall u \sqsubseteq v) P(u,v)$ for $(\forall u)(u \sqsubseteq v \implies P(u,v)),$ and so on for $\exists,$ and for $\in$ as well.
Then Hrbacek's GRIST is the following condition on $\sqsubseteq,$ with four axiom schemata:
R elativization condition on $\sqsubseteq$: $\sqsubseteq$ is a total dense preordering with minimal element
    $\emptyset$ and no maximal element; i.e. the conjunction of 


*

*Partial ordering: $(\forall u,v,w)((v\sqsubseteq u \wedge w \sqsubseteq v) \implies w \sqsubseteq u) \wedge u \sqsubseteq u;$

*Preordering: $(\forall u,v)(u \sqsubseteq v \wedge v \sqsubseteq u);$

*Minimality of $\emptyset$: $(\forall u)(\emptyset \sqsubseteq u);$

*Illimitability: $(\forall u) (\exists v) (u \sqsubset v);$

*Density: $(\forall u,v) (u \sqsubset v \implies (\exists w)(u \sqsubset w \sqsubset v)).$
Axiom schemata, in which we use words so as not to have our eyes completely glaze over: 
For any well-formed formula $P$ of $\mathscr{HB}$ depending on finitely many variables,


*

*T ransfer: for all $u \sqsubseteq v$ and $x_1,\ldots, x_n \sqsubseteq u,$

$P^u(x_1,\ldots,x_n) \iff P^v(x_1,\ldots,x_n)$


*S tandardization: for all $u \sqsupset \emptyset$ and for all 
$A, x_1, \ldots, x_n,$ there are $v \sqsubset u$ and $B \sqsubset v$ such that, for
every $w$ with $v \sqsupseteq w \sqsupset u,$

$(\forall y \sqsubseteq w)(y \in B \iff y \in A \wedge P^w(y,x_1,\ldots,x_n)).$


*I dealization: For all $A \sqsubset v$ and all $x_1,\ldots,x_n,$

$(\forall a \in 2^A_{\mathrm{fin}})\big([a\sqsubset v \implies (\exists y)(\forall x \in a) P^v(x,y,x_1,\ldots,x_n)]
   \iff[(\exists y)(\forall x \in A)[x \sqsubset u \implies P^v(x,y,x_1,\ldots,x_n)]\big).$


*G ranularity: For all $x_1,\ldots,x_k,$ if $(\exists u) P^u(x_1,\ldots,x_k),$ then

$(\exists u)[P^u(x_1,\ldots,x_k) \wedge (\forall v)(v \sqsubset u \implies \neg P^v(x_1,\ldots,x_n))]$

