The following seems, to me, to be "the nicest theorem that does not have a special name". It is a wonderful blend of Topology, Geometry and Analysis. Moreover, it has a short and simple statement, involving only the notions of Euler characteristic and of a Lie group.

Theorem: The Euler characteristic of a connected nontrivial Lie group is zero!

The proof is also simple, being an application of Lefschetz fixed point theorem: Since the left translation endomorphism of a Lie group $G$ (by a non-trivial element) has no fixed points, Lefschetz numbers are homotopy invariant, and the Lefschetz number of the identity map is the Euler characteristic of $G$, the result follows (note that compactness of the Lie group is not really an issue).

Of course, the result (and the nice short proof) is known, but I think it should be much better known, and part of a lot of standard books.