jmc

Since $f(2)=6$, we have $f^{-1}(6)=2$. (Note that the hypothesis that $f$ has an inverse implies that there are no other values of $x$ with $f(x)=6$.) Similarly, $f(1)=2$ implies $f^{-1}(2)=1$. So $f^{-1}(f^{-1}(6))=f^{-1}(2)=\boxed{1}$.