Games don't need stories.

[Riachu]

Kenryoku_Maxis said:
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.

Examples of games like that would be The Longest Journey, Dreamfall: The Longest Jorney, the Syberia series, Indigo Prophecy, Heavy Rain, Snatcher, Policenauts, Flower Sun and Rain, etc.  Final Fantasy and Metal Gear Solid arguably count as well.