Do we know of any differentiable function whose derivative is not an elementary function? This may be a silly question, but in the light of this answer, as pointed in the comments, finding an example may be pedagogical.
More importantly, can we prove the existence or non-existence of such a function?
Edit: An answer that is not in the form $f(x)=\int_0^x g(x)$ would be much appreciated. The point is to find an example that would be of value for the above answer.