A fundamental domain for the modular group can be viewed a subset of the upper half plane $\mathbb H$ that contains exactly one point from every orbit of the $SL(2,\mathbb Z)$ action on $\mathbb H$. This can be defined analogously for subgroups $G$ of $SL(2,\mathbb Z)$. The notion carries over to modular forms and modular functions for $G$: If $G$ is a genus zero group the orbits of the images of $G$ under its Hauptmodul can be studied, and the two notions should agree.
For functions $f$ which are not modular under any subgroup of $SL(2,\mathbb Z)$ (such as the square root $\sqrt{E_4}$ of the Eisenstein series) such a region could be defined analogously: A minimal domain such that $f$ does not assume the same value for two distinct points inside that domain, but still assumes all possible values. Is such a concept studied in the literature? I would be very grateful for any hints. Thanks a lot!
Edit: Let us consider the example $f: \mathbb H \to \mathbb C$, $\tau\mapsto \frac{E_6(\tau)}{\sqrt{E_4(\tau)}^{3}}$ with $E_4$ and $E_6$ the Eisenstein series of weight $4$ and $6$. The function $f$ has an integer $q$-expansion $f(\tau)=1-864q+269568q^2+\mathcal O(q^3)$ with $q=e^{2\pi i\tau}$, so it is holomorphic and invariant under $T:\tau\to \tau+1$. It has formally weight $0$. Under the $S$-transformation restricted to the imaginary axis $\tau=i \beta$, $\beta>0$, we have that $f(i/\beta)=-f(i \beta)$ since $E_6$ picks up a minus sign while there is none for $\sqrt{E_4}$ (this is unambiguous since $(i\beta)^4$ is real). So $f$ is anti-invariant under $S$, which does enlarge the $SL(2,\mathbb Z)$ fundamental domain in such a way that a region around $0$ is also included in it. It seems to me that the associativity of the group action of $SL(2,\mathbb Z)$ on $\mathbb H$ does not carry over to such functions. It would be really interesting to understand this better!