jmc

To solve the equation $f(f(x))=4,$ we first find the values $x$ such that $f(x)=4.$

Either $f(x)=-x+3$ (for $x\le 0$) or $f(x)=2x-5$ (for $x>0$). If $-x+3=4,$ then $x=-1.$ Note that this value satisfies $x\le 0.$ If $2x-5=4,$ then $x=9/2.$ Note that this value satisfies $x>0.$ Therefore, the solutions to $f(x)=4$ are $x=-1$ and $x=9/2.$

Next, we solve for the values $x$ such that $f(x)=-1.$ If $-x+3=-1,$ then $x=4.$ This value does not satisfy $x\le 0.$ If $2x-5=-1,$ then $x=2.$ This value satisfies $x>0.$

Finally, we solve for the values $x$ such that $f(x)=9/2.$ If $-x+3=9/2,$ then $x=-3/2.$ This value satisfies $x\le 0.$ If $2x-5=9/2,$ then $x=19/4.$ This value satisfies $x>0.$

Therefore, the equation $f(f(x))=4$ has the solutions $x=2,$ $-3/2,$ and $19/4,$ for a total of $\boxed{3}$ solutions.