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.

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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.

The introduction to Kontsevich's thesis Intersection theory on the moduli space of curves and the matrix Airy function gives references to the solution to the problem for closed surfaces. Are there any other papers in the literature that discuss this problem?