For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We show the same kind of maximal entropy rigidity holds for a class of metric measure spaces known by now as $RCD^*(K,N)$ spaces. While the upper bound follows quickly, the rigidity case is quite involved due to the lack of a smooth structure in $RCD^*$ spaces.