In my opinion, the best games out there are the ones with amazing story lines and interesting complex characters, that take you on an emotional journey as you live vicariously through their lives. 
Near the top of the tree for best game stories, has to be The Last of Us 1&2, so to coincide with the release of the new Last of Us TV show, here's is me chatting script about both the show and the games, as well as a few other honourable mentions for games with fantastic story lines.