It seems likely to me that the (graphs of) Hecke operators on the self-product of a modular curve have this property. This might be a little hard to verify because of the cusps, so it might be better to work with suitable Shimura curves (quotients of the upper half plane by a torsion free arithmetic subgroup of an indefinite rational quaternion algebra).
This gives curves C of genus > 1 and infinitely many curves Gamma_i in C \times C such that both projection maps from Gamma_i to C are finite and etale. This shows that the self intersection of each Gamma_i is negative. The degrees of these maps go to infinity, hence so does the genus of the Gamma_i.