Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.