Recall that $\exp(1,x) = e^x$ and $\exp(n+1,x) = e^{\exp(n,x)}$.
Recall that $f(x)$ is transexponential if $f(x)$ is eventually greater than $\exp(n,x)$ $\forall n \in \mathbb{N}$
I am looking for a (general) reference on these types of functions (or any paper about these functions, or maybe even a few pages of a textbook).
Note: I have tagged model theory (and now logic) since the only context in which I have encountered transexponential functions is in relation to Wilkie's Conjecture (and so model theorists know about these functions). Please note that I am looking for a reference about transexponential function in general, and not a link to an exposition of Wilkie's Conjecture.
Note 2: I have added a bounty to this question. I am trying to get my hands dirty with transexponential functions from $\mathbb{R}^+ \to \mathbb{R}^+$. The most helpful answer would be one where I could "in some sense" compute the derivative (locally). Please do not answer with "piecewise continuous segments" + bump functions.