I'd probably say that I enjoy games that cover the adventure genre. So, I'd place games like Shenmue I & II, Heavy Rain, Beyond: Two Souls, Shadow of Memories, etc as my favourites. I'm not really a gamer so to speak, but a lover of cinema and narrative. Collecting swords and building stats, like most RPGs, don't really interest me. They're a nussance that I put up with in order to try and flesh out the storyline. I guess that's why whenever I complete a game, it still says I have only collected 5% of the trophy achievements. =P

Personally, I'd prefer individual stories per game, as I like to have a solid conclusion. There is enough creativity in the world to invent something new, instead of flogging a dead horse till it is nothing but a pile of bones and scrag-end.