Even though the title of this question pretty much captures what I'd like to know, I'll add two side questions:

1) Is it difficult to get a handle on the totality of functions that arise if one drops the growth condition?

2) Presumably these functions have no known number theoretic interest; is there a theoretical reason to explain why they shouldn't?

Edit: In light of the early responses (thank you all!), particularly Scott Carnahan's, I'm particularly interested to know about modular functions with essential singularities and I suppose non-zero weight. As Scott points out, for quite arbitrary $g$, $g(f(z))$ will be $\Gamma$-invariant if $f$ is $\Gamma$-invariant ($\Gamma\subset SL_2{\Bbb Z}$). Then I suppose one could multiply such a $g(f(z))$ by a classical modular function of non-zero weight.

Does this observation admit some sort of converse, something in the spirit of the Weierstrass factorization theorem for entire functions, say?