jmc

The solutions to $f(x)=6$ are $x=-2$ and $x=1,$ so if $f(f(x))=6,$ then $f(x)=-2$ or $f(x)=1.$

The line $y=-2$ intersects the graph of $y=f(x)$ twice, so the equation $f(x)=-2$ has two solutions.

The line $y=1$ intersects the graph of $y=f(x)$ four times, so the equation $f(x)=-2$ has four solutions.

This gives us a total of $2+4=\boxed{6}$ solutions.