jmc

As shown by the graph, there are $3$ values of $x$ for which $f(x)=3$: when $x=-3$, $1$, or $5$. If $f(f(x))=3$, it follows then that $f(x)=-3,1,5$. There are no values of $x$ such that $f(x)=-3$. There is exactly one value of $x$ such that $f(x)=1$ and $5$, namely $x=-1$ and $3$, respectively. Thus, there are $\boxed{2}$ possible values of $x$.