jmc

We start from the inside out. Since $2<4$, $f(2)=2^2=4$. So $f(f(f2)))=f(f(4))$. Since $4\le 4$, $f(4)=4^2=16$. So $f(f(4)=f(16)$. Since $16>4$, $f(16)=\sqrt{16}=\boxed{4}$.