This thread is a perfect example of how difficult it is to convey exactly what a person intends through a post. It also provides insight into argument, particularly in the realization that making assumptions about even the definition of words such as [a] story can completely negate the persuasive power of one's words.
That being said, I wold define a "story" as something that makes events cohesive. If you will agree with me in what a story's function is, then I ask you to define a story by it's function, and to thus give it definition.
Under this definition, everything has a story because everything can be said to happen for some reason, be it by logical deduction or something as dissident as divine intervention.
For example, the game of checkers has a story because each players assumes a competitive role against an opponent of another color. Every move in the game acts as a reaction to the initial story, or the taking of roles, of the players.
On the other end of the spectrum, let's say you have issues with registration at the airport and miss your flight. The plane crashes. Are these events linked? An issues with registration are completely unrelated to navigation of the airplane, yet someone may argue that you missed your flight for a reason. Therefore, this situation has a "story".
Every game has story, or some cohesiveness, if the person taking part chooses so. By redefining the definition of a story though, this can be negated. This argument is based in opinion, essentially.
My initial "feeling" is that games are always art. I probably feel this because, by making something purposely devoid of art, doesn't it become art because it's saying something about art by not being art (paradox)? Therefore, the only thing that's not art is anything done without purpose. Isn't that what being human is about, though? I don't think we ever "just do" anything, or, if we do "just do" anything, it's probably so insignificant in comparison to the other things done with purpose that we don't care about it.
'Course, we could just be chemical computers with arms and legs...