Limbo, Papers Please, I Am Bread, Please Don't Touch Anything, are all good examples of games I would also consider art. These games are genuinely different and many of them are considered incredibly enjoyable by most people.
If you want something which really is just art and not much of a game, The Cat And The Coup is free on Steam. Frog Fractions 2 is also free and you can find it using Google (you have to play the whole game to understand).
For the most part I think gamers and developers are a bit sensitive about their medium, trying to argue it's "art" because it has a positive connotation. I think they should just enjoy it for what it is, as most games are designed to be fun instead of something else. That said, I think VR might try to change this. A friend of mine at Oculus mentioned how it's the very early stages, comparing VR to movies from ~100 years ago. Back then they just filmed plays, as they didn't realize what they could do with the medium. I'm sure VR will have lots of typical blockbusters following generic formulas, but I think indie developers (of which there are more every day) will build some pretty incredibly stuff in the coming years.