- Hajime no Ippo (Fighting Spirit): It's an anime about boxing, but even if you're not into that you should really watch it. A young bullied boy meets a pro boxer and aspires to be stronger. He goes on to become a professional himself through the course of the series. The story is great, and even the filler eps and movies are greatly entertaining.
- Death Note: A Death God drops his Notebook into the human world. If a name is written in the notebook, that person dies. A young student finds this book and starts punishing criminals. It's got great intellectual stuff. A GREAT story. You should watch it. Preferably right now.
- Code Geass: Futuristic anime; Britannia has invaded and overtaken Japan and most of the world. Some rebels remain, and a small group starts fighting back for Japan's freedom. Also a REALLY nice anime, with some nice plot twists and intellectual stuff as well.
- Hikaru no Go: About the game of Go (sort of like chess); A spirit from ancient Japan accompanies a young boy on a "quest" to become a professional Go player. It sounds like a dull concept, but it's a really nice story.
- Chobits: It's a bit more silly than the others, but very "slice of life" with not too much action stuff and such. It's set in a near future, with very human looking robots with fully sentient AI called Persocom's. People use them as computers, phones, etc. For cooking, cleaning and work. Some fall in love with them and get married, though that is mostly frowned upon. A student finds a mystical persocom dumped nearby, and the story revolves around those two. It's got a nice story, and I like it a lot.
There's a few that I could think of right now. Hope you enjoy some of them