Solution:
We have $(f \circ f \circ f \circ f \circ f)(x) = \ln (\ln (\ln (\ln (\ln (x)))))$.

The domain of this function must satisfy $\ln (\ln (\ln (\ln (x)))) > 0$, which implies that $\ln (\ln (\ln (x))) > 1$, $\ln (\ln (x)) > e$, $\ln (x) > e^{e}$, and $x > e^{e^{e}}$.