First I'm going to recommend Professor Layton. It's basically puzzle after puzzle, but it's very much about the story and atmosphere. I don't know if you've ever seen a Studio Ghibli movie - Kiki's Delivery Service, Princess Mononoke, Spirited Away, etc - but it's kind of like jumping into one of those, wandering around. I definitely definitely recommend headphones for this one.
Next up is Rune Factory. This game has a dickload to do. It's based off of Harvest Moon, and just in case you don't know what that means, I'll explain. Harvest Moon is a series of farming games - yes, farming - where you grow crops, buy livestock, and so on. But it also has a social sim aspect, where interacting with characters and remembering their birthdays and giving them gifts will make them like you more. You can even marry a girl, once you get her to like you enough. So take all of that, and stuff an RPG into it. That's Rune Factory. There's also other things to do, like fishing, mining, cooking, crafting, reorganizing and decorating your home, various holidays and contests that happen over the course of the game, and stuff I can't even remember. The combat is realtime, and you can choose from a wide variety of weapons and magic. If you're into that sort of thing, this game can keep you occupied for quite a while.
Next up is Devil Survivor Overclocked. This one is a little more hardcore than most people around here prefer, so don't pick this one up unless you're looking for a challenge. It's a tactical RPG, which if you don't know what that is, basically means it's like an RPG board game, like Fire Emblem. The whole map is a grid, and your characters can move around on the spaces on that grid. Each character acts as a team leader for two demons, all of whom can gain experience and level up through combat. The story is, you and your friends get stuck in a massive government quarantine, along with thousands of other people. This was a result of demons starting to pop up, and the government can't risk them spreading to the rest of the world. You and your friends find a way to recruit demons to fight the more dangerous ones, and your goal is to save everybody and end the lockdown. This game has the longest playtime of any on my system, and the main campaign alone lasted me 90 hours - before the extra content after the credits roll. Definitely worth checking out, especially with the sequel coming out soon.
And another hardcore one, is Shin Megami Tensei IV. You play as a newly recruited samurai, tasked with keeping people safe from the demons that escape from Naraku - a dungeon which the people are forbidden from entering. However, when a being known as the Black Samurai begins corrupting people, turning them into demons, and causing widespread chaos, you are forced to enter Naraku, to capture the Black Samurai, and discover the secrets of the dungeon. I don't want to tell you more about the story, because it has a ton of plot twists. It's another game where you fight with demons - it's made by the same company as Devil Survivor. Except this is a more traditional turn-based RPG, and is in a 3D game world. In the game, there are certain mechanics such as demon fusion - the act of fusing two demons to make a stronger one - and the various interaction options that make this game really cool. For example, during a battle, you don't need to fight. You can actually talk to the enemy, and do things like get items and money, recruit them, or if they're at a much higher level, you can negotiate a ceasefire. All of these things give you experience, so you don't have to just kill one thing after the other to level up, making grinding pretty much obsolete. You can also get experience from the various side quests in the game. There's plenty to do here, and it's an 80 hour campaign, and it's pretty much my favorite game on the system.
Let me know if you want more details about anything.