Below, by "nice structure" I mean "o-minimal expansion of $(\mathbb{R};<)$ by countably many continuous functions and open relations."
Suppose $\mathfrak{A}=(\mathbb{R};<,...)$ is a nice structure in a language $\Sigma$. Letting $U$ be a unary predicate symbol not in $\Sigma$, say that an $\mathfrak{A}$-ordinator is a function $F:C\rightarrow\omega_1$ for some club $C\subseteq\omega_1$ such that there is some $\Sigma[U]$-formula $\varphi(x)$ with the property that for every $\alpha\in C$, if $A\subseteq\mathbb{R}$ has ordertype $\alpha$ then $$\operatorname{otp}(\varphi^{\mathfrak{A},U\mapsto A})=F(\alpha).$$
We can compare $\mathfrak{A}$-ordinators by setting $F\trianglelefteq G$ iff $F(\alpha)\le G(\alpha)$ for club-many $\alpha$; since $\mathfrak{A}$ is required to be reasonably simple, $\mathfrak{A}$-ordinators are always comparable with respect to $\trianglelefteq$. Let $O(\mathfrak{A})$ be the ordertype of the set of $\mathfrak{A}$-ordinators with respect to $\trianglelefteq$. For example, $(\mathbb{R};<,+,1)$ has only one nonconstant ordinator, namely the identity map, essentially due to the fact that we can't "compress" unbounded sets of reals; on the other hand, it's easy to see that $O(\mathbb{R}; <,+,\cdot)$ is much greater. For example, here is a formula $\varphi$ "representing" the map $\alpha\mapsto\alpha+1$ (described somewhat intuitively):
$\varphi(x)$ holds iff each of the following holds:
If $U=\emptyset$ then $x=0$.
If $U$ has a maximal element then $x\in U\vee x=\max(U)+1$.
If $U$ is nonempty and bounded above but has no maximum then $x\in U\vee x=\sup(U)$.
If $U$ is not bounded above then $x=1$ or $x\in \widehat{U}$, where $$\widehat{U}=\left\{-1-{1\over u}: u\in U_{<0}\right\}\cup\left\{1-{1\over u}: u\in U_{>0}\right\}.$$
I believe I have arguments showing that $O(\mathfrak{A})$ is always closed under composition and that we can find nice structures with ordinals arbitrarily high below $\omega_1$, but they're each a bit tedious. Nope, both arguments broke down!
Moreover, I can't answer the following hopefully-simple test question:
Question: What is $O(\mathbb{R};<,+,\times)$?
(I'm actually more interested in comparing $O(\mathbb{R};<,+,\times)$ and $O(\mathbb{R};<,+,\times, \mathit{exp})$ but that seems much harder.)
It may also be worth considering modifying the definition of $\mathfrak{A}$-ordinator to require $A$ to be bounded above, in order to make $O(\mathfrak{A})$ (call this version "$O_b(\mathfrak{A})$") potentially interesting even when $\mathfrak{A}$ is quite weak. But since my question is about a sufficiently strong structure, this variation won't matter here.
