jmc

Inputting $f^{-1}(x)$ into $f$, we have $f(f^{-1}(x))=4f^{-1}(x)+5$, so $x=4f^{-1}(x)+5$. Solving this equation for $f^{-1}(x)$, we get that $f^{-1}(x)=\frac{x-5}{4}$. Thus, we have $$\begin{array}{rl}{f}^{-1}(f^{-1}(9))& =f^{-1}\left(\frac{9-5}{4}\right)\\ & =f^{-1}(1)\\ & =\frac{1-5}{4}\\ & =\boxed{-1}.\end{array}$$