Value of $a+a^a+a^{a^a}+a^{a^{a^a}}+... $

Question

Is there any closed form expression of $a+a^a+a^{a^a}+a^{a^{a^a}}+...$ up to $n$ terms? I am mostly interested in $a=2$

My try:

$a+a^a+a^{a^a}+a^{a^{a^a}}+... =a(1+a^{a-1}+a^{{a^a}-1}+a^{{a^{a^a}}-1}+... )$, but I do not know how to continue.

Answer 1

With $a=2$, the first terms make the sequence $$\{2,6,22,65558,\cdots\}$$ The problem is that the next one is $$\sim 2.0035\times 10^{19728}$$

Notice that, just by itself, $$2^{65558} \sim 8.4034\times 10^{19724}$$