Is there a context free, non-regular language $L$, for which $L^*$ is regular?
$L = \{a^n b^n \mid n\in\mathbb{N}\}$ 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.