Maximal degree extension needed to define an isogeny over a finite field

The Honda-Tate theorem classifies abelian varieties over finite fields up to isogenies. We use this theorem to answer the following question: if two abelian varieties over $\mathbb{F}_q$ of dimension $g$ become isogenous over some extension of $\mathbb{F}_q$, but not over any proper subfield, how large can the degree of this extension be (in terms of $g$)? We give an application to prove the existence of infinitely many geometrically nonisogenous elliptic curves defined over any infinite algebraic extension of $\mathbb{F}_q$.