jmc

We approach this problem by working from the inside out, so we first find $f(17)$. Since $17\ge 7$, $f(17)=17-13=4$. Next, $4<7$, so $f(f(17))=f(4)=(4{)}^2-4=12$. Finally we have that since $12\ge 7$, $f(f(f(17)))=f(12)=12-13=\boxed{-1}$.