In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)

I want to emphasize a problem which comes from mathematical physics which is unsolved which is indicating that we don't understanding what triangulation is. And the problem is extremely simple. You take a manifold, and you just ask how many triangulations it has with a given number of simplices. So you have your manifold $X$ and you have the number of triangulations with k simplices $N_k(X)$ and you want to know what happens to it as $k$ goes to infinity, roughly. You take triangulations up to isomorphism. It's bounded below by $(1 > + \epsilon)^k$ and it's bounded from above by $k^k$, roughly. That's kind of trivial, you just keep subdividing and you see how many automorphisms you have an a $k$ element set. The question is, where is the truth? And nothing is known, just absolutely blank. For surfaces you know, it's like that [exponential] and physicists kind of made that computation.

[...]

The whole point is that the manifold must be fixed, if you vary the manifold you would have $k^k$.

The subtle point is fix a topological manifold, how does the combinatorics tell you something about the topology. We think we understand it, but when we do this computation we don't. There's absolutely not a direct link between the two. We have a zero level question in topology, we cannot answer it.

What papers are there in the literature that discuss this problem?