Transexponential Functions 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.  
 A: A way to obtain transexponential functions while having some control on their growth is through the Abel equation for the exponential function. This is the following equation in $E$: 
$\ \ \ \ \ \ \ \ \ \ \ \ $ $\forall r\gg1,E(r+1)=\exp(E(r))$ $\ \ \ \ \ \ \ \ \ \ \ \ $(A)
(where $\forall r\gg 1$ means: for sufficiently large $r \in \mathbb{R}$).
Any solution $E:(a,+\infty) \rightarrow \mathbb{R}$ to (A) which is continuous is transexponential. Indeed, fix $n \in \mathbb{N}$ and write $m:= \min E([a+1,a+2))=E(b)$. The function $k \mapsto \exp(k,m)$ eventually dominates $k \mapsto \exp(n,b+k+1)$, say starting from $k_0 \in \mathbb{N}$. For $r >b+ k_0$, we have $E(r)\geq E(b+\left\lfloor r-b\right\rfloor)=\exp(\left\lfloor r-b\right\rfloor,m)>\exp(n,b+\left\lfloor r-b\right\rfloor+1)>\exp(n,r)$.
Hellmuth Kneser first came up with quasi-analytic solutions to this equation and derived some bounds on the derivative. If you can read German, see his article.
You can find complements on this topic in the Appendix A of this Phd thesis (only the introduction is in French, and it is translated into English in Appendix B).
