SquiddyCracker
Banned
If there will also be stories from GAF implemented in the Inn, someone could tell the tale of the singer Dennis the Duck (or Dennis Dye Acke), who spread songs everywhere in the country about 'Goose and Fox' and it's visitors being responsible for everything bad in the world, until he was finally banished and sank with his ship the 'pars quinque'.
Maybe something more fleshed out, but you get the idea.
Expanding on Lore's idea, occasionally the guards should throw out a particular NPC every now and then who admits to the party he'll just return under a different alias to piss on the boorish politicals for shits and giggles.
A quest would involve Ameerox - a powerful and well-known arbiter of the den - who is revealed to be secretly using powerful magicks which alter the memories of past conversations for hidden and insidious ends! The party could choose to side with the dastardly Ameerox, thus gaining free tips or whatever mechanics the GAF Inn would possess at a discount. To reveal his treachery would mean they can get some special "arbiter" privileges at the Inn, which include throwing out NPCs and guiding topics of the conversations.
I don't think it's a good idea to base any quests or storylines on any Gaffer (aside from Evilore, who owns the site and most likely will continue to do so) who might end up perma-banned or simply forgotten in a couple of years.
Just general "gafferisms" should be fine though, like two men having a loud disagreement on a political matter before both are kicked out of the inn. Or a patron complaining about the brothel in the city, and how the whores there are below his standards.