Asymptotic analysis of Skolem's exponential functions

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^x}$. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type $omega$. As a consequence we obtain, for each positive integer $n$, an upper bound for the fragment below $2^{n^x}$. We deduce an epsilon-zero upper bound for the fragment below $2^{x^x}$, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.