In the space of Maass forms (weight 0, level 1 for simplicity) there are the cusp forms and also the (real analytic) Eisenstein series. None others come to mind, but is the space of Maass forms simply the direct sum of cusps forms and Eisenstein series?
This is true for modular forms, but for Maass forms the constant coefficient has two complex parameters, and so it seems like in general you can't cancel out the constant term of a general Maass form with just the Eisenstein series.