Can something finite over $\mathbb{C}(q)$ be a modular form?

A non-constant modular form has a natural boundary on the real line. A power series that's algebraic in $q = e^{2\pi i \tau}$ can't.

In characteristic $p>0$ the quotient field ($K$ say) of the ring generated by $E_4,E_6$ (i.e. modular forms of level one) is of transcendence degree $1$. But $q$ (in the sense of the Tate curve) is transcendental over $K$ (I proved this in J. Number Theory 58 (1996) 55-59). So any non constant modular form or modular function is transcendental over $\mathbb{F}_p(q)$.