As it was shown by Igor, there is no univeral bound on number of such geodesics. Let me show that the number can not be infinite.
Assume it is possible to get infinite number of such geodesics.
From analyticity, we get one-parameter family of such geodesics and we can pass to its analytic extension $\gamma_\tau$.
Note that the geodesics in the family stay simple and disjoint locally. Globaly it only may happen that $\gamma_0=\gamma_c$ for some parameter $c>0$. In this case the surface is a torus.