jmc

Working from the inside out, since $2<4$ we have that $f(2)=(2{)}^2-1=3$. Continuing, since $3<4$ we have that $f(f(2))=f(3)=(3{)}^2-1=8$. Finally, since $8\ge 4$ we have that $f(f(f(2)))=f(8)=3(8)-2=\boxed{22}$.