It is well known that many real valued real functions are not Riemann integrable on subsets of $\mathbb{R}$, but formally an antiderivative may still exist. May I see an example of a function having no antiderivative? Or does any function (without additional hypothesis) always have an antiderivative?

Edit: Does a continuous function always satisfy Choquet's characterization? If yes, take $f$ differentiable and such that $f'$ is bijective. In order to explicitly compute the antiderivative of $f'^{-1}$ I find myself needing that $f'$ is also differentiable. Could such an additional hypothesis be ruled out somehow?