Solution:
$\boxed{\pi}$

Since $f(x)$ never equals $\cos(1)$ for $x \in (0, \pi)$ but $f(0) = \cos(1)$, the period is at least $\pi$. However, $\cos(x+\pi) = -\cos(x)$, so $\cos(\cos(x+\pi)) = \cos(\cos(x))$.