I'm trying to find a book or a paper on infinite exponentiation: more precisely, it should be proving its full interval of convergence on the positive real line, i.e. if $x\in\mathbb{R}^+$, then $x^{x^{x^\cdots}}$ converges if and only if $e^{-e}\le x\le e^{1/e}$.
While I didn't manage to find any books about it, there are these two papers from The American Mathematical Monthly:
Also, there's this paper on arXiv:
Every one of them is problematic.
Paper #1 doesn't make sense towards the end of page 242 and is rather incomprehensible there.
Paper #2 shows only the convergence for $1\le x\le e^{1/e}$ and the conclusion is
DEFINITION. For all real $x$ such that $0\le x\le e^{1/e}$, $x^{x^{x^\cdots}}$ is the real solution for $t$ of the equation $x^t=t$, and in case this equation has two solutions, then the lesser one.
That is not satisfactory at all.
Paper #3 makes an incorrect assumption on page 15, which is the following:
If the first first derivative is $|z'(y_1^*)|\lt 1$ (that is $-1\le z'(y_1^*)\lt 0$...
There shouldn't be $-1\le$ but $-1\lt$, the author probably does so to avoid the analysis of the case when $x=e^{-e}$, but at the same time he further in the text states that the infinite tetration for $x=e^{-e}$ is convergent. So that one is incomplete.
In Ioannis Galidakis' A Collection of References For Infinite Exponentials and Tetration, see items #7, 19, 88, 110, 111, 166. See also On the Convergence of Iterated Exponentiation—I by Creutz/Sternheimer (1980) and On ${x^{x^{x^{{\cdot}^{{\cdot}^{\cdot}}}}}}$ by Louis A. Talman (written in 1999, I think).