Be prepared to spend quite a lot of time on these.
(The first two require Flash; the third is a standalone 3D game, but probably requires significant processor power.)
First up: the lovely and mesmerizing flOw. [Link updated in 2019; the original link to the downloadable Flash game no longer works, so I’m now linking to the company’s page about the game.] You control a creature that looks a little like a microbe, swimming around, shifting up and down through twenty levels/layers, eating food, avoiding or attacking other creatures, growing, evolving.
The game was apparently created as part of Jenova Chen's MFA thesis, but you don't need any of the theory to play the game. All you need to do at first is move your mouse pointer around; your creature moves toward the pointer.
Later, you'll learn that holding down the button causes the creature to move faster; that eating dots makes the creature grow longer (in an interesting and neat way, with sound and visual effects); that eating the flower-like creature with the red dot causes you to descend one layer in the world (it gets darker as you descend), while eating the one with the blue dot causes you to ascend one layer; you'll find more complex creatures like your own that you can eat, and later even more complex ones that turn orange before attacking you; you'll discover that eating one kind of thing causes your mouth to temporarily grow wide, while eating another causes a segment of your body to evolve and eventually grow fins. And there's lots more.
All of this with pretty ambient music in the background, and a totally gorgeous (and coherent) visual design.
There are some hints in the forum, including instructions on how to enter cheat mode.
Next: Line Rider. Draw a line, then watch as a little person on a sled rides along it. Play with it for a little while; then go watch some Line Rider videos, especially the astonishing Jagged Peak Adventure, to get a sense of what's possible. Then play with it some more. (You will appreciate the videos more after you've spent at least a couple minutes playing with it yourself.)
I never managed to get it to complete a full loop; my hand may not be steady enough to be successful with this until there's an eraser. Still, mighty cool.
Finally: Lugaru, a 3D-graphics game that you download and run locally (not in your browser). In some ways it's kind of the opposite of the above games: it's a very goal-oriented combat game. You control an anthropomorphic rabbit in combat fatigues; you fight other rabbits, and wolves, and such; there are a whole bunch of key combinations that let you jump and spin and kick and roll and run and do fancy aerial acrobatics and use various weapons and so on. The graphics are smooth, the moves nifty, and if I had any aptitude for this kind of game I would probably be playing it still; as it was, I barely made it through the training/tutorial sequence. Available for Mac, Windows, and Linux. The demo version is free; you can register (which I think lets you use weapons) for $20.