jmc

First, we find all $x$ such that $f(x)=2$ by drawing the line $y=2$ and finding the intersection points.



Thus, $f(x)=2$ for $x=-3$, $x=0$, and $x=3$. So, if $f(f(x))=2$, then $f(x)=-3$ ,$f(x)=0$, or $f(x)=3$.

Since $f(x)\ge 0$ for all $x$ ,the equation $f(x)=-3$ has no solutions.

We see that $f(x)=0$ for $x=-4$.

And the graphs of $y=f(x)$ and $y=3$ intersect at $x=-2$, and once between $x=3$ and $x=4$ at the red dot. This means the equation $f(x)=3$ has two solutions.



Therefore, the equation $f(f(x))=2$ has a total of $\boxed{3}$ solutions.