  An elliptic orbifold is the quotient of an elliptic curve by a finite group.
Eskin and Okounkov proved that generating functions for the number of branched
covers of an elliptic curve with specified ramification are quasimodular forms
for the full modular group $SL_2(\mathbb{Z})$. They later generalized this
theorem to the enumeration of branched covers of a pillowcase, i.e. the
quotient of an elliptic curve by the elliptic involution, proving
quasi-modularity for $\Gamma_1(2)$. We generalize their work to the quotient of
an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving
quasi-modularity for level $\Gamma_1(N)$.
  One corollary is that certain generating functions of hexagon, square, and
triangle tilings of compact surfaces are quasi-modular. These tilings enumerate
lattice points in moduli spaces of flat surfaces. We analyze the asymptotic
behavior as the number of tiles goes to infinity, theoretically giving an
algorithm to compute the Masur-Veech volumes of moduli spaces of cubic,
quartic, and sextic differentials. We also deduce that the volume is polynomial
in $\pi$.
