Another game (for the PS2) that I love is Katamari Damacy. It's very simplistic, very stylized, and definitely not a typical American game. I'll just cut and paste from the wikipedia article to describe the story:
The game's plot concerns a tiny prince on a mission to rebuild the stars, constellations and Moon, which his father, the King of All Cosmos, has playfully destroyed. This is achieved by rolling a magical, highly adhesive ball called a katamari around various locations, collecting increasingly larger objects, ranging from thumbtacks to schoolchildren to mountains, until the ball has grown large enough to become a star. The game falls under both the puzzle and action game genres, since strategy as well as dexterity are needed to complete a mission.
You just use both the analog sticks to move the prince around collecting the objects. It's a great nonviolent game for kids, but it falls into the Japanese-absurdist-LSD-inspired categories, and some of the cutscenes and intros need some explaining.
I have not played any of the sequels, but hopefully they don't stray too far from what made the original so unique.