It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F_2$, the free group on two generators, cannot be embedded as a closed subgroup of a locally compact amenable group. However, by a result of Pestov, $F_2$ embeds as a closed subgroup of $\mathrm{Aut}(\Bbb Q,\leq)$ and the latter group is (extremely) amenable.

Are there topological groups that cannot be embedded as a closed subgroup of any amenable group?