Is there a context free, non-regular language $L$, for which $L^*$ is regular?

Question

I know that there are non-regular languages, so that $L^*$ is regular, but all examples I can find are context-sensitive but not context free.

In case there are none how do you prove it?

Answer 1

$L = \{a^n b^n \mid n\in\mathbb{N}\}$ is context-free but not regular (classical example). So is $L' = \{a^n b^n \mid n\in\mathbb{N}\} \cup \{a,b\}$.

$L'^\ast = \{a,b\}^\ast$ is regular.