  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.
