  We determine the size of spaces of higher order Maass forms of even weight
for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth
at the cusps is allowed, the spaces of Maass forms of a given order are as
large as algebraic constrictions allow. We show the analogous statement for the
spaces of holomorphic forms. For functions on the universal covering group of
PSL(2,R) we introduce the concept of generalized weight. For the resulting
spaces of higher order Maass forms with even generalized weight we show that
the size is maximal.
