'Bog' is an old term for the jakes, the WC, the stable Tom Crapper's mount, where you keep the porcelain throne. it's the necessary room, the powder room, the backhouse, the latrine.

For some reason Amos is going to a party in a Las Vegas outhouse. I suppose it's a multi-holer since there are usually more than a few people at a party, unless it's one of mine. In any case, he's sure to get stinkin'.