  Given a countable residually finite group, we construct a compact group K and
two elements w and u of K with the following properties: The group generated by
w and the cube of u is amenable, the group generated by w and u contains a copy
of the given group, and these two groups are dense in K. By combining it with a
construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda
and formation of diagonal products, we construct an example for which the
latter dense group is non-exact. Our proof employs approximations in the space
of marked groups of LEF ("Locally Embeddable into Finite groups") groups.
