Give an example of a non-regular language $L$ such that $L^*$ is regular

Question

I can't think of an example of a non-regular language $L$ such that $L^*$ is regular. . Any help ?

Answer 1

Define $L=\{a^nb^n|n\in \mathbb N\}\cup \{a,b\}$

It's not hard to see that while $L$ is not regular, $L^*=\Sigma^*$.