Games don't need stories.

[Kenryoku_Maxis]

Plenty of games can have lackluster stories and be good games. But that doesn't mean games (as a whole) don't need stories. Some games focus their entire concept around plot and character interaction (aka a story) and just tie in concepts like gameplay and player interaction on top of the narrative.

So the short counter-example to the opening post is that not all games need stories, but some games do.