What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

Your definition of $g(x)$ does not make sense for $x! \gt e^{(1/e)} \approx 1.44467$ because it does not converge as seen in the answer to this question and this question. It is less than this in about the ranges $-4.970 \lt x \lt -4.103$ and $-0.380 \lt x \lt 1.614$. Within those ranges, you are doing fine.

I would say that everything looks fine, as long as you define that $x! = \Gamma(x+1)$.