Consider the equation: $f(x) = \ln (f(x))$

Does there exist a function that satisfies this equation? It is obvious that $f(x)$, if it exists, is not differentiable. Taking the derivative of both sides would yield $f'(x) = \dfrac{f'(x)}{f(x)}$. But, then you could cancel out the $f'(x)$ terms and you find that $f(x)=1$, which is a contradiction. Therefore, $f(x)$ could not be differentiable (if it exists at all).

It could also be written as the limit to the sequence of functions $f_0(x) = x$ and $f_n(x) = \ln \left( f_{n-1}(x) \right)$ should the limit function exist.

Never mind. I figured it out. No, the function does not exist. Proof:

The domain of $f_n(x)$ would be $^ne$ where the superscript before the number refers to tetration. So, $^2e = e^e$, $^3e = e^{e^e}$. But then the limit domain would not exist.