Q:

Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(x)$ is ultimately greater than $\exp^N(x)$. That is there is we can find $r\in \mathbb{R}$ such that for all $x>k$ we have $\exp^N(x) < f(x)$ where  $\exp^N(x)$ stands for $\exp(\exp(\cdots \exp(x)\cdots))$ $N$ times. 
Is it possible to find an o-minimal expansion (in the model theoretic sense) of $\bar{\mathbb{R}}$ where an $f$ as above is definable?

A:

This is a well-known open problem, see e.g. http://www1.maths.leeds.ac.uk/maloa/lecturenotes/lyon.pdf . All currently known o-minimal expansions of the reals (such as the pfaffian closure of $\mathbb R$) are exponentially bounded.

