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?

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?