OK, I'll bite - belatedly. Of the platforms listed, I have 4 of these, so I'll limit my suggestions to the area where I know things.
Grand Theft Auto III, Vice City, San Andreas, and related titles - unless you have them on PC. These were also on the original XBox but I don't know about compatibility with XBox 360. GTA3 is one of the first real "sandbox" games and if you can get past the violent premise of the series, it's a lot of fun. There are some spinoffs of this series on the PSP, some or all of which have been ported back to PS2.
Okami - avoid the Nintendo Wii port of this game, buy on the PS2, and be aware that some odd humor from the Japanese made it through to the US cut of this game
Ratchet & Clank - There are three R&C platform games (the original game w/o subtitle, R&C2 "Going Commando", and R&C3 "Up Your Arsenal"). I have these, and they're all excellent. (Note that I'm listing the US titles here, the game names varied in some international territories.) Note also there is one additional title called "Ratchet: Deadlocked" - which is not a platform game and is missing "Clank" in the title. There are some spinoffs of this series on the PSP as well, and at least one of the PSP games has been ported back to PS2.
The original Katamari Damacy and sequel, We <Heart> Katamari, are on PS2. Sequels are on the PSP (Me & My Katamari), XBox 360 (Beautiful Katamari), and PS3 (Katamari Forever).
Kindgom Hearts, 1, 2, and Chain of Memories. Various other additions to the series are on the PSP (Birth by Sleep), and Nintendo DS (see below). There are even cell phone games in the series.
The original Burnout games were on PS2, I'm actually looking for an inexpensive copy of Burnout Anthology... these are lots of fun.
There was a special edition of Space Channel 5 for the PS2, you can get it for about $15.
If you did not get Psychonauts on the PC, you might find it for the PS2. Edit: A friend tells me that the PS2 version has some "crippling bugs" so I'm guessing maaaaybe you'd be better off with the PC version.
ICO is the game that predated the highly rated Shadow of the Colossus, both on PS2... the third and final chapter in the series is slated to be released on the PS3.
The sequels to Xenogears, the Xenosaga series (3 games), were on PS2.
Finally, the PS2 also had all three Jak & Daxter platform games. The 2nd and 3rd were a sharp and strange departure from the first game - the series goes from bright and colorful to dark and depressing.
Grand Theft Auto IV - also available on PC and XBox 360
Ratchet & Clank Future - there are two retail titles ("Tools of Destruction" and "A Crack in Time"), and one PSN chapter in the middle, "Quest for Booty". Tools of Destruction was an alright game but didn't seem as innovative as the original series on the PS2 - but you need to play through it for the story which leads into Quest for Booty and then Crack in Time, which is awesome.
The Rock Band series and Guitar Hero 3 and later are great on the PS3, although most people get these on their XBox 360 because more people have XBox 360s. Rock Band: Green Day just came out recently, and Rock Band 3 is coming this winter. Rock Band: The Beatles is a trip.
The Sly Collection is coming out on PS3 in November - so if you were wondering why I didn't mention the Sly Cooper series on the PS2, now you know why.
Little Big Planet - a sandbox platform game where you can create and share your own levels. The sequel, Little Big Planet 2 is coming out in November and is supposed to be backwards compatible with the content from LBP.
Burnout Paradise is on PS3 and XBox360 and on the PC.
3D Dot Game Heroes is a wonderful tongue-in-cheek homage to the Legend of Zelda series, and is only available on the PS3.
Uncharted and Uncharted 2 are PS3 exclusives that are awesome if you like the Tomb Raider / Indiana Jones type of adventure/story/FPS games. Great story and visuals, not to hard to play, and good (useful) unlock-able achievements.
The Last Guardian, the last chapter of the series with ICO and Shadow of the Colossus, is supposed to come out on the PS3.
Kingdom Hearts III is rumored to eventually come out on the PS3 (est. 2012).
Katamari Forver is in some ways a port of the original Katamari Damacy 1 and 2 games to the PS3 - in addition to some new levels, if you've played the original games you'll see many of their levels re-used here (and you can play with either 'classic' mode or with Forever's new mechanics, including the ability to jump).
The original God of War games (1+2) are updated and released as God of War Collection for the PS3, and God of War 3 is out now too. This series is exclusive to the PS2/3; there are some cell phone and PSP games in the series as well.
Scott Pilgrim vs The World: The Game is now exclusively available on PSN and is supposed to come out for XBox Live for XBox 360 after few weeks.
Fat Princess is a PS3 PSN exclusive.
Thexder Neo, recently mentioned, is on the PS3 and I think PSP as well.
Heavy Rain is exclusive on the PS3 (some have compared it to Alan Wake which is exclusive to the XBox 360).
Remember that your PS3 (as well as your PS2) can play any original PSX titles. So that will include many of the Final Fantasy series games, as well as Chrono Cross and Xenogears.
The Legend of Zelda: Twilight Princess
Super Mario Galaxy and its recently released sequel, Super Mario Galaxy 2
Zelda: Skyward Sword is coming out probably sometime next year.
Boom Blox and its sequels are fun.
Rayman: Raving Rabbids is silly and fun as well - it's a series of mini-games so you can play in short bursts.
Nights was resurrected on the Wii - the play control is not great but if you were fond of the original Saturn game, you might pick this up
I'd probably be shot if I didn't mention Mario Kart Wii.
Flips Twisted World is supposed to be a Super Mario Galaxy-like game from Majesco, it's coming out in September.
Epic Mickey is coming out for the Wii in November.
A trippy reworking of the original Super Mario Brothers, Super Mario Bros. Wii, came out earlier this year.
Donkey Kong Country Returns has been announced for the Wii.
Wii Fit Plus - combine your video games with exercise.
Remember that your Wii can also play GameCube games, such as Zelda: Wind Waker, Super Mario Sunshine, and the classic horror game (that you have to play through 4 times to reach the final ending), Eternal Darkness.
Don't forget that you can download (for a nominal fee) many classic SNES and Nintendo 64 titles for your Wii as well.
Zelda: Phantom Hourglass and Zelda: Spirit Tracks.
Professor Layton - two games are out in this series now, and the third (P.L. and the Unwound Future) ships in the US in September. In Japan, there is even a full length feature film with Professor Layton. Mind bending puzzles, and a great plot.
Chrono Trigger - in case you never played it on SNES or on the original Playstation. (To play the sequel, Chrono Cross, you'll need to buy it in the PSX format and play it on your PS2 or PS3.)
Early next year, Okamiden, the sequel to Okami, will be coming out on the DS.
Kingdom Hearts: 358/2 Days is available only on the DS, and Kingdom Hearts: ReCoded is going to be released on the DS as well.
If you like the Mario+Luigi RPGs (which began back on the SNES with Super Mario RPG: Legend of the Seven Stars, if I recall correctly), these continue on the DS with Mario & Luigi: Bowser's Inside Story.
Your Nintendo DS may also (depending on model) be able to play any Game Boy Advance cartridge, which would include Mario & Luigi Superstar Saga, Metroid Fusion, and the GBA port of Zelda: A Link to the Past which expands the game with some additional content.