If you listen to only two recommendations in this thread, let it be Valkyria Chronicles and Demon's Souls. Two experiences that stand out like few other games this or the last generation can. Especially Demon's Soul, which is a game challenging for all the right reasons and as long as you are alert, you'll make it through. This is a game where if you die, it's because you messed up.
Infamous, a rather nice exclusive open city game where you get to play as a super human with electricity. Comic style all the way with good/evil choices to make.
For other great games. LittleBigPLanet for make your own platformer stages, soon LBP2 for make your own any type of game, and ModNation Racers for create your own race tracks. All fun with a lot of downloads of custom stages, many quite well done.
3D Dot Game Heroes is a throw back to the old NES open world adventure games a la Legend of Zelda or Golden Axe Warrior. Only here it's on top of amusing references to retro games, in 3D with pixels, and multiple swords that can do quite a bit. Even hitting people with a giant fish is practical.
Katamari Forever, if you missed out on the original two Katamari games on PS2. Roll things up, crazy, strange, addictive.
Critter Crunch, a rainbow vomit jewel of a puzzle (that's a compliment). It has everything from fast paced twitch action of Money Drop and Dr. Mario with clever puzzles you must think out and will test you quite well. Cheap title too.
Fat Princess and Castle Crashers are out that anyone can get into and play. Online multiplayer, strategy, and amusing style for both.
Joe Danger fun title to get for that time trial/high score type of game of old. Very easy to get into, very easy to control and explain. It's a gem.
Any of the PixelJunk games are fun and easy to get into.
Shatter is Breakout/Araknoid if the blocks fought back and tried to kill you. Lots of items, special blocks, boss fights makes this rather fun and clever. Though I would recommend getting this only on PSN if you can't get it on PC.
Seriously get Demon's Souls. It will make you remember what it felt like to beat a game boss and feel the joy of success of your gaming skills.