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 Apologies if I am out-of-date; updates welcomed.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!