jmc

Working from the inside out, we first compute $f(4)=4^2-4(4)-1=-1$. Next we find $f(-1)=(-1{)}^2=1$, and then $f(1)=1^2=1$. Putting these together, we have $f(f(f(f(f(4)))))=f(f(f(f(-1))))=f(f(f(1)))=f(f(1))=f(1)=\boxed{1}$.