This question is closely related to this one. Belatedly, it occurred to me that I'd probably picked the wrong test question. I'm leaving that question up because I don't think it's bad per se, but I think this one is what I should have asked:
Suppose $\mathfrak{A}=(\mathbb{R};<,\ldots)$ is an o-minimal structure in a countable language $\Sigma$. Letting $U$ be a unary predicate symbol not in $\Sigma$, say that an $\mathfrak{A}$-ordinator is a function $F:\omega_1\rightarrow\omega_1$ such that there is some $\Sigma[U]$-formula $\varphi(x)$ such that for club-many $\alpha<\omega_1$, if $A\subseteq\mathbb{R}$ has ordertype $\alpha$, then $$\operatorname{otp}(\varphi^{\mathfrak{A},U\mapsto A})=F(\alpha).$$
(Note that this differs from the definition in the above-linked post; that post worked "club-many" into the specific question, rather than the definition.)
It's easy to show that there are $(\mathbb{R};<,+,\times)$-ordinators which are not $(\mathbb{R};<,+)$-ordinators (basically, the only non-constant example of the latter is the identity function). I'm curious whether we can keep improving; in particular, whether adding the function $e^x$ gives us any power in this sense.
Question: Are there $(\mathbb{R};<,+,\times,\exp)$-ordinators which are not $(\mathbb{R};<,+,\times)$-ordinators?
I'm not quite sure where to look for a positive example; each map $\alpha\mapsto\alpha^n$ is an $(\mathbb{R};<,+,\times)$-ordinator, but I don't see that $\alpha\mapsto\alpha^\omega$ is an $(\mathbb{R};<,+,\times,\exp)$-ordinator. So the first natural function which might be a candidate for escaping one structure isn't obviously accessible by the other.
