For the following either give a proof or exhibit a counterexample:
Let $\gcd (a, n) = 1$ then
$$ a^{\phi(\phi(n))} ≡ 1 \pmod n $$
Consider $a=3, n=5$, which means $\gcd(3,5)=1$.
We have $\phi(5)=4$, and so $\phi(\phi(5)) = \phi(4) = 2$.
And so
$$ \begin{align} a^{\phi(\phi(n))} & \equiv 3^{2} \pmod 5 \\ \\ & \equiv 9 \pmod 5 \\ \\ & \equiv 4 \pmod 5 \\ \\ a^{\phi(\phi(n))} & \not \equiv 1 \pmod n \end{align}$$
This is a counter-example to the given proposition.