I'm not an expert, but recently I read the beautiful paper by Ono "The last words of a genius" on the Notices of the AMS, Dcember 2010, which is related to your question.

Let $M \colon \mathbb{H} \to \mathbb{C}$ be a *smooth* function that transforms like a weight $k$ modular form and such that $\Delta_k(M)=0$. Then we say that $M$ is a *weight $k$ harmonic Maass form*.

Any harmonic Maass form can be uniquely written as

$M=M^{+} + M^{-}$,

where $M^+$ is the *holomorphic part* and $M^-$ is the *non-holomorphic part*. Then the modular forms are exactly those harmonic Maass forms such that $M^-=0$. 

In the general case, the holomorphic part of a harmonic Maass form is *not* a modular forms, but it is still a very interesting object. For instance, when $k=1/2$ it is a so-called *mock theta function*.