jmc

First, by definition of the inverse of a function, $f(f^{-1}(19))=19$, so $f^{-1}(f(f^{-1}(19)))=f^{-1}(19)$.

We then find the inverse of $f(x)$. Substituting $f^{-1}(x)$ into the expression for $f$, and noting that $f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$, we get that $$x=(f^{-1}(x){)}^3-8.$$Solving this equation for $f^{-1}(x)$, we get that $f^{-1}(x)=\sqrt[3]{x+8}$. Then $f^{-1}(19)=\sqrt[3]{19+8}=\sqrt[3]{27}=\boxed{3}$.