  We investigate compositional iteration of fractional order for transseries.
For any large positive transseries $T$ of exponentiality 0, there is a family
$T^{[s]}$ indexed by real numbers $s$ corresponding to teration of order $s$.
It is based on Abel's Equation. We also investigate the question of whether
there is a family $T^{[s]}$ all sharing a single support set. A subset of the
transseries of exponentiality 0 is divided into three classes ("shallow",
"moderate" and "deep") with different properties related to fractional
iteration.

Rings and Algebras;Classical Analysis and ODEs