Find a convergent solution for $a$

Question

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you extend your range (term used loosely) of possible answers to $a\in \Bbb{C}$? A rigorous proof is nice but ultimately not necessary to answer this question.

Answer 1

The sum cannot converge, even for complex $a$. If an infinite series converges, the summands must tend to $0$. However, since each summand is the factorial of the previous summand, then the summands would also tend to $0!=1$, a contradiction.