The number of triangulations of a set of $n$ planar points: Why so difficult? After hearing Emo Welzl speak on the subject this summer, I know the number of 
of triangulations of a set of $n$ points in the plane is somewhere between
about $\Omega(8.48^n)$ and $O(30^n)$. Apologies if I am out-of-date; updates welcomed.

I mentioned this in class, and wanted to follow up with brief, sage remarks to give 
students a sense for (a) why it has proved so difficult to nail down this quantity, 
and (b) why so many care to nail it down.  I found I did not have adequate answers
to illuminate either issue; so much for my sageness!

I'd appreciate your take on these admittedly vague questions.  Thanks!
Here's one more "applied" reason why we care about triangulations. There's a body of work on mesh compression where the goal is to use as few bits per vertex as possible to encode a mesh (mainly to aid in storage and transmission). The particular base of the exponent in the number of triangulations of a planar point set provides an information-theoretic lower bound on the number of bits needed per vertex (specifically, $8.48^n$ triangulations means you need at least 8.48 bits per vertex). Such bounds can then be compared with actual mesh compression schemes to determine their efficacy.