All .hack games? G.U. included?
Seeing some RPG in there, I'd put out Persona 3 FES (has to be, must be) and Persona 4 (though this is getting a Vita remake, if you have that console). Turn-based RPG/dating sim etc.
I don't really play much else PS2 games, but if you don't mind dabbling around since it's not costing you any money, Ar Tonelico is sure to take up a lot of time.
I found Star Ocean Til the End of Time fun, thought it had interesting combat mechanics for its time.
I believe only the gen 1 (the super fat ones with gloss finish) are backwards compatible with PS2. I'm not sure about the gen 3 (the newest ones), but my gen 2 (matte finish) definitely can't play PS2.
Or was it that Sony removed backwards compatibility with an update? I can't remember.
I think I have the 120gb model? The one that was like, $249.99 I think. I honestly don't think you really need a lot unless you count on downloading a lot of movies and games right onto the hard drive, 120gb is covering enough for me.
As for games, if you enjoyed Guilty Gear as a fighting game, they're releasing the reboot(?) of AC+ as AC+R in November on PSN, so I guess that's something to look forward to. If you like fighting games in general, you could wait for BlazBlue Chronophantasma, and... I would say Persona 4 Arena, but apparently you live in Europe. Pity.
Outside of fighting games, Borderlands 2 has been providing a lot of mileage of play and has splitscreen co-op for your brother if you want.
If you like really quirky RPGs that don't exactly have a lot of substance in their story (outside of fanservice and hurrdurr humor), Neptunia is fun. The first one has a really lackluster combat system, I believe people compared it to Trinity Universe? The second one is a lot better from what I understand (still need to play through it, OTL).
If you can play more patient RPGs and pour a lot of time into it, I hear Rune Factory and Atelier series are good. I can't say for myself because I have the worst attention span despite owning both, barely even put time into them.
If you like music games that are just artwork (because honestly, that's what PS3 games mostly are), Child of Eden is beautiful. Its replay value ultimately boils down to whether you want to unlock more extra crap or beat your high score or whatever, but it's pretty and centers around Genki Rockets as the music, so...
Pretty much whatever game I have that seems somewhat relevant, the rest of my games are meh. OTL