How about more than the current four visibility levels? Probably the most frequent use case for this would be to allow group members to apply for membership in sub-groups without having to make those sub-groups public. So let’s say you have a group for all members in an association, then those members (and only those) should be able to see various thematic working groups or what have you, either as a matter of transparency or in order to make it possible for them to join those group. Likewise, those in the board group would be able to see various sub-committee groups etc.
If I’m not mistaken, currently the only way to allow users to apply for membership in a group is to make the group public.