Starships Unlimited. A cool 4X game where you have to keep advancing technologically, build a small fleet of ships for exploring, fighting, and taking over worlds, and deal with other races as you see fit by force or diplomacy. You can trade technologies with them as well, which can be quite helpful.
This one's a real gem that I've returned to many times, with a good bit of depth to it. Forget MOO, I'd pick this one instead. Has a bit of a learning curve though.
Not originally an independent game as such, but I highly recommend
The Ur-Quan Masters if you've never played the classic Star Control II before. The Ur-Quan Masters is a remake of it for modern computers, and is completely free. This classic is a mix of different game styles.
Strange Adventures in Infinite Space is perfect for very short but unique games which you can win or lose within 20 minutes. They have a sequel out, but I haven't tried it yet.
Huh, all three of my recommendations are sci-fi stuff, go figure. I've played a bunch of independent stuff, those three are just the ones that come to me off the top of my head.
I'd recommend some roguelikes such as the wide variety of Angband variants, but those have a very large learning curve for all the varied keyboard commands alone, not to mention making it very far without dying. Fun stuff for some people though, myself included.