Pryms are geometrically degenerate. Let $f:Y\to X$ be an étale double cover of a genus $g>1$ curve $X$. Up to taking isogenous quotients, which will not affect "geometric degeneracy", the Prym of $f$ is the kernel of $\text{Alb}_f : \text{Alb}_Y\to \text{Alb}_X$, where the domain is an Abelian variety of dimension $2g-1$, and where the target is an Abelian variety of dimension $g$. The Prym has dimension $g-1$, so codimension $g$ in $\text{Alb}_Y$. So now let $C\subset \text{Alb}_X$ be any curve that does not contain the origin. Then the inverse image $\text{Alb}_f^{-1}(C)$ is a subvariety of $\text{Alb}_Y$ of dimension $g$ that is disjoint from the Prym.