Apostrophes of possession: key question is 'Who is the owner?' (or who are the owners?). Then 1) write down the owner(s), 2) add an apostrophe, 3) add s if you need it for the sound. This method cuts out complicated rules about plurals.
There are no exceptions, but occasionally two correct options. The book belonging to James can be James' book or James's book depending on how you would say it. But THE APOSTROPHE IS ALWAYS PLACED IMMEDIATELY AFTER THE OWNER(S). Lucy's dress, children's clothes, several boys' coats etc.
Games room needs no apostrophe; the games don't own the room.