jmc

We start from the inside and work our way out: $$f(-1)=(-1{)}^2-2(-1)=3.$$ Therefore $$f(f(f(f(f(f(-1))))))=f(f(f(f(f(3)))).$$ Now $f(3)=3^2-2\cdot 3=3$. We can use that fact repeatedly to conclude $$\begin{array}{rl}f(f(f(f(f(f(-1))))))& =f(f(f(f(f(3))))\\ & =f(f(f(f(3)))\\ & ⋮\\ & =f(3)=\boxed{3}.\end{array}$$