Is a kind of reverse Kleene star of a context-free language context-free?
Let me solve a simpler question first. Suppose we'd like to know


  If $C$ is context-free, must $F(C)=\{w: ww \in C\}$ be context-free?


The answer is no: $F(\{a^i b^i c^j a^j b^k c^k\})=\{a^i b^i c^i\}$.



As for the original problem, consider the context-free language that contains words of the form $\{a^i b^i c^j d a^j b^k c^k d\}$ or those which don't contain exactly two $d$'s.

Define $D=\{w: w^\ast \subseteq C\}$. Suppose $D$ is context-free. Then $E=D\cap a^\ast b^\ast c^\ast d$ is context-free.

I claim $E=\{a^i b^i c^i d\}$, which is not context-free. Suppose $w = a^i b^j c^k d$. We have $w \in E$ exactly if $w^n \in C$ for every $n$. This condition is clearly satisfied when $n \neq 2$, because $w^n$ will not have exactly two $d$'s. For $n=2$, $w w=a^i b^j c^k d a^i b^j c^k d\in C$ which is equivalent to $i=j=k$.