For any $g >0$ there exists an abelian surface $A$ containing a smooth curve $C$ of genus $g$; the surface can be assumed to be simple if $g > 1$:
For $g=1$, one can just consider $C \times C$. For $g>1$, consider the generic abelian surface with a polarisation of type $(1, g-1)$. By the adjunction formula, a smooth curve in the linear system corresponding to the polarisation will have genus $g$. (Such abelian surfaces can be constructed by considering suitable quotients of principally polarised ones ($n=1$))
One gets abelian varieties of any dimension $>1$ by taking products.
However, this does not produce a fixed abelian surface containing curves of all genera $g>1$ but it seems likely that such surfaces should exist. It would be interesting to know whether this is possible for the generic principally polarised surface, or more generally ppav of any dimension $n$.