On ordered groups of regular growth rates

We introduce an elementary class of linearly ordered groups, called growth order groups, encompassing certain groups under composition of formal series (e.g. transseries) as well as certain groups ${\mathcal{G}}_{\mathcal{M}}$ of infinitely large germs at infinity of unary functions definable in an o-minimal structure $\mathcal{M}$. We study the algebraic structure of growth order groups and give methods for constructing examples. We show that if $\mathcal{M}$ expands the real ordered field and germs in ${\mathcal{G}}_{\mathcal{M}}$ are levelled in the sense of Marker & Miller, then ${\mathcal{G}}_{\mathcal{M}}$ is a growth order group.