I am not sure, if you are still looking for an example, and whether you feel that my examples are artificial.

Every locally compact, almost connected group is a projective limit of Lie groups, in particular a projective limit of metrizable groups. The limit becomes metrizable itself, if and only if the limit is countable. There is a book by Hofmann and Morris about pro-Liegroups, which studies projective limits of Lie groups.

Also the vector space $C_b^\infty(X) = \cap_k C_b^k(X)$ of smooth bounded functions is a projective limit of normed spaces.