When trying to evaluate$$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$I noticed that the answer was bound to be complex for any $x$.
Plugging in a very, very large real number in for $x$ will eventually become complex, and the $\log$ of a complex number is always complex or imaginary.
But what happens when we try to evaluate $$\lim_{x\to\infty}\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$
We see that $$\lim_{x\to\infty}\ln(x)=\infty$$Meaning that if we try to do this repeatedly, we should still get infinite as our final result.
But we also see that for arbitrarily large $x$ where $x$ is real and finite, the result is a complex answer.
Which creates a sort of contradiction. What does it evaluate to?!?!