  We prove that the upper metric mean dimension of $C^0$-generic
homeomorphisms, acting on a compact smooth boundaryless manifold with dimension
greater than one, coincides with the dimension of the manifold. In the case of
continuous interval maps we also show that each level set for the metric mean
dimension is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$
with the uniform topology.
