Yes!

I was going to write about the exact same games. The YAKUZA games (especially 2) and GUN are great... and GUN is here on GOG as well

Was going to say Mercenaries: Playground of Destruction but then I remembered that pretty much any GTA3 like game ran like molasses on those era machines.

I didn't play a lot of games on PS2, possibly only ever completed two, but one of them kind of fits, as it was Jak and Dexter: The Precursor Legacy. Wasn't completely open, but enough to be a lot of fun. (The other game was Baldur's Gate: Dark Alliance which is extremely linear.)

same , tried out some in emulator , i liked one i dont remember its name:D it was a tactical game with little team on a grid map

hmm Beyond Good and Evil is on ps2 , thats a good game at least the pc version
same for Prince of Persia: The Sands of Time

I can only speak for the original Xbox (I still buy games for mine), but I'm pretty sure all of these was on the Playstation too.

The first Mercenaries was a pretty fun game.

Beyond Good & Evil is one of the most original games I've played. I must have finished it five or six times total by now. Great world, gameplay, soundtrack and characters. One of my all-time favorites.

Scarface: The World Is Yours. It didn't get very positive reviews back then, but I had a lot of fun with it. The driving mechanics and open world gameplay was solid enough, and it had a great 80s soundtrack. I remember all the side activities for extra income, like smuggling drugs in a boat from the islands, doing races, managing the different fronts.. I loved it. And yes, I have a poster of the movie on my wall.

I haven't played BGE yet, although I've bought it on GOG, so i'm not sure - if you can travel between various places, without needing to do them in a certain order, like in the Elder Scrolls Games, or having interconnected areas you can choose to go like in Might and Magic 6-8 or like Baldur's Gate 1, those are a open world games, for example. :) Did the first two games have anything censored out? I remember hearing that the later games had minigames and the host club deleted when it was brought over.

Actually, I'm pretty sure that the earlier Might and Magic games (1-5) are also open world (though Swords of Xeen is less open than the others).

(If we allow PS1 games, then there's one interesting question: Is Final Fantasy Origins open world?)

Never played GameCube games, but from the PS2, 3 open-worldish games that stood out for me were Naruto Ultimate Ninja 5, Shadow of the Colossus and Final Fantasy XII. Not sure if I'd consider either of them as better than the other 2, tho.

Your loss. Tons and tons of amazing games on both systems. Amazing games like Eternal Darkness (probably the single most creative, insane, amazing, shocking and fantastic actual Psychological Horror game ever made, those who experienced it can't praise it enough for a good reason) or Metal Gear Solid 3 blow most everything nowadays out of the water.

I've always had a soft spot for GTA Liberty City Stories (I like mob stories) and also the Jack & Dexter and Ratchet & Clank games. There was always something you missed and had to go back and look around for later.

I miss the good old Burnouts. Paradise was fine, but nothing beats well-designed, closed-circuit tracks. Not to mention that the whole series died after that. Thanks, EA.

Addendum: The Need for Speed: Hot Pursuit games were superb as well. Can you tell I like arcade racers? Too bad the whole genre seems to have mostly died out.