Question:
Let $f(x) = \sqrt{x}$ and $g(x) = x^2.$ Find $f(g(f(g(f(8))))).$

Answer:
We can evaluate this the hard way, or we can see that $g(f(8)) = (\sqrt{8})^2 = 8$. Therefore, $f(g(f(g(f(8))))) = f(g(f(8))) = f(8) = \sqrt{8} = \boxed{2\sqrt{2}}.$