  We describe the additive structure of the graded ring $\widetilde{M}_*$ of
quasimodular forms over any discrete and cocompact group $\Gamma \subset
\rm{PSL}(2, \RM).$ We show that this ring is never finitely generated. We
calculate the exact number of new generators in each weight $k$. This number is
constant for $k$ sufficiently large and equals $\dim_{\CM}(I / I \cap
\widetilde{I}^2),$ where $I$ and $\widetilde{I}$ are the ideals of modular
forms and quasimodular forms, respectively, of positive weight. We show that
$\widetilde{M}_*$ is contained in some finitely generated ring
$\widetilde{R}_*$ of meromorphic quasimodular forms with $\dim \widetilde{R}_k
= O(k^2),$ i.e. the same order of growth as $\widetilde{M}_*.$
