The limit exists for the first two examples that come to mind, namely topological entropy on the full shift and on certain simple Markov shifts.

If $X \subset \Sigma_d^+ = \{1,2, \dots, d\}^{\mathbb{N}}$ and $\sigma$ is the shift map, then for the topological entropy the quantity $a_n$ denotes the number of words of length $n$ that appear in some sequence $x\in X$. If $X$ is the full shift, then $a_n = d^n$, the entropy is $h = \log d$, and we quickly see that $a_n / e^{nh} = 1$ for all $n$.

Slightly more interesting is when you have a Markov shift, say $X\subset \Sigma_2^+$ determined by the transition matrix $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$.

In this case it's not hard to show that the sequence $a_n$ is actually the Fibonacci sequence, and thus writing $\phi = \frac {1+\sqrt 5}2$ and $\psi = \frac { 1-\sqrt 5}2$, we have
$$
a_n = \frac 1{\sqrt 5} (\phi^{n+2} - \psi^{n+2}).
$$
Since $|\psi|<1$, this shows that the limit of $a_n / e^{nh}$ exists.

My guess is that a similar argument works for other Markov shifts and shows that the limit exists in those cases, based on obtaining a recurrence relation for $a_n$ and then an exact formula using standard tools for solving such sequences.

All that said, it's not immediately clear what the significance of the limit is, and I don't know of any name for it. For other interesting shifts, such as sofic shifts or shifts with specification, I'd be surprised if the limit always exists.

What **is** certainly quite important is to have conditions under which the ratio $a_n / e^{nh}$ is bounded away from $0$ and $\infty$. Such estimates are a significant part of arguments on the uniqueness of a measure of maximal entropy (and more generally uniqueness of equilibrium states), in particular the proof that such a measure satisfies a Gibbs property. For example, see Bowen's 1975 paper "Some systems with unique equilibrium states". (Dan Thompson and I also struggled with this not too long ago in Section 5.1 of this paper.)

It turns out that in the general setting, one of those bounds is immediate -- the sequence $a_n$ is submultiplicative, and so it's not hard to show that $a_n \geq e^{nh}$ for all $n$, whatever other properties the shift space has. Getting an upper bound on $a_n / e^{nh}$ is harder and requires some sort of specification property.