Everblue (PS2)

Although i'd never heard of this game before a few days ago, it was apparently popular enough to get a sequel, as well as two spiritual sequels in the Endless Ocean series for Wii (which I also haven't heard of, though I admit the Wii is one of my blind spots).
Anyway, the game's abut diving and looking for treasure on the ocean floor. There's also some plot regarding an ancient civilisation or something, but it doesn't come into things until much later in the game. Other than that, the game's setting is mostly pretty realistic, with only very minor fantasy elements.
For most of the early game, you'll be walking the ocean floor, tapping your sonar button until you get a hit on some treasure, then closing in and taking that treasure. Then you take it back to the surface to evaluate and sell it or give it away to someone who needs/wants it. you should also be careful, as you can sell items needed to complete optional side quests, and even progress in the story. So only sell items that you have more than one of. Eventually you'll start finding shipwrecks, which you explore with full 3D movement, and you'll look for treasure inside those too.
The second shipwreck, the Juno is harder to fin than the first, since you won't be given directions until you beat a guy in an underwater swimming race. This race is the worst part of the game. It's too hard, boring, doesn't fit in with the feel of the rest of the game, and comes accompanied with some awful background music. Luckily, the ship is actually there whether you get the directions or not, and you might even be able to find a handy map on some cool internet blog if you look hard enough.
Although the game itself is actually quite repetitive, it never becomes boring. Finding a bunch of mysterious items and finding out what they all are is pretty cool, and the game has a great atmosphere too. Inbetween dives all your quests, shopping and item appraisals take place in the two small towns on the island. These bits add a lot to the feel of the game: they're always sunny, the people are all friendly, and the whole experience just feels really warm and happy. Interestingly, references to things like romance, alcohol, death and so on, that often get removed in English translations of videogames are left intact here. (And by death, I mean in the sense of an old woman missing her late husband, and so on, rather than the usual kind of videogame deaths that only serve as catalysts for more conflict.)
Another noteworthy aspect of the game is the complete lack of conflict. Even the guy who challenes you to a race does so only out of friendly, time-killing rivalry. There's no enemies or hostile creatures underwater, no evil regime to overthrow on land. It's actually quite refreshing to play a game that's so completely positive!
I thouroughly recommend you play this game if you have the chance.