  Given a compact smooth boundaryless manifold with dimension greater than one
endowed with a locally positive non-atomic measure $\mu$, we prove that typical
$\mu$-preserving homeomorphisms have upper metric mean dimension, with respect
to the Riemannian distance, equal to the dimension of the manifold. Moreover,
we prove that $\mu$ is a measure of maximal metric mean dimension, with respect
to the variational principle established in [VV17].
