Crackdown - All about being a high-flying, super-powered beast of man. The best part about the game is getting your guy powered up and just leaping from one building to the next and raining down missiles on the city. Story and missions could have been better, but they weren't enough to stop me from having a blast with this game.
Test Drive Unlimited - Grab a car and cruise the idyllic Hawaiian countryside. It's not as deep as other racers out there, but it's got a nice amount of variety in the races and a lot of different cars and motorcycles. More than any other racers out there, it's good for relaxing and taking things easy.
Saints Row 1 and 2 - All about going out in the city and causing mayhem and having fun. These are the games that GTAIV should have been. Story? Yeah, it's there, but it's not really the point of the game. Lots of mission and sub-mission variety and few things that get in the way of just having fun.
Dead Rising - Got to the mall, kill zombies. One of the most visceral and thrilling sandbox experiences out there. A huge amount of weapon variety, lots of cool little secret areas to find, and load and loads of the walking dead to maim, dismember, blow up, run over, and otherwise kill. Respawning escaped inmates with Humvee and minigun was pretty annoying and some of the rescue missions bordered on impossible, but, hey, that's because this a man's sandbox game.
Mercenaries 2 - Shoot guys, run over stuff, and blow up buildings. That's pretty much all this game is about. There's a story there, but it really doesn't matter. All you need to know is that somebody wants you to cause mayhem. Has some nice mission variety and some descent building destruction.
Red Faction: Guerrilla - Basically, this is Saints Row: Mars. A lot of the missions are similar and there's a big emphasis on fun and mayhem. One of the cool additions is that buildings can be blasted to pieces in a way that looks cooler than in any other game out there. Plus, it's scifi, so that instantly makes it more awesome than SR.
Far Cry 2 - Jamming guns and malaria, that's what this game is about. FC2 takes notes from the book that Volition wrote on sandbox games. Again, there's a story here, but it really doesn't matter that much. This one's all about mayhem and more mayhem. Also, it's fun to set fires and let them burn up guys that are trying to attack you or waste a jeep with a rocket launcher. There's more of an emphasis on tactics and stealth here, as running and gunning can get you killed pretty quickly.
Stalker: Shadow of Chernobyl - One part Fallout, one part FEAR, and...uh...maybe some other stuff thrown in for extra flavor. Somewhat like an FPS, when you first start the game you're about as effective in combat as a newborn babe. But once you reach the end of the game you can tear everybody in Pripyat a new sphincter. This game absolutely DRIPS atmosphere, even if it is a bit janky in that Eastern-European way. Lots of weapon variety and the shadowing looks pretty good when cranked all the way up.
Fallout 3 - By far the sandbox experience of this generation. Mutants, ghouls, raiders, despotic government-types, and radiation set against a half-destroyed Washington DC and its suburbs. You can be a wasteland hero or a backstabbing jerk, it's all up to you. Which ever path you choose, it's a great experience.