Situations where Kleene star of A is context-free, but A is not
Let $L$ be any non-context-free language, then we can construct the language $A = \Sigma\cup L$ - i.e. $A$ is either an element of $\Sigma$ or a string from $L$ (these can happily overlap too). This is clearly not context free either (if it were, then we would have a PDA for  it, and we can easily modify this PDA to recognise $L$).

If we now take $A^{\ast}$ however, we get $\Sigma^{\ast}$, which is most certainly context free.