We know the Stone-Čech compactification $(h, \beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,\gamma X$) is any other Hausdorff compactification, then there is a continuous map $f:\beta X \rightarrow \gamma X$ such that $fh=k$.
What about if I restrict to considering just all metrizable Hausdorff compactifications? To fix ideas, say $X$ is a noncompact metric space. (In particular $\beta X$ will not be metrizable.) Is there a maximal metrizable Hausdorff compactification of $X$?
I am guessing no: if we're given any metrizable Hausdorff compactification of noncompact metric space $X$, we can make another metrizable Hausdorff compactification that is larger (in the sense of the first paragraph above).