In terms of manga, I of course have read the classic Azumanga Daioh (and own it) in its entirety. My mother, when visiting South Korea on a business trip, ended up buying me a hentai light novel translated into Korean without knowing it had pornographic images inside it (thanks mom....?). Hah. I enjoy Sayonara Zetsubou Sensei and its satirical content. Its description of the meaning on the universe was rather beautiful (and included a small chibi of Kim Jong-il, labeled "Dear Leader," he's an inescapable truth of the universe!)
I have some Japanese Zelda manga, which is pretty neat but not exactly a good example of a manga having a deep story. Today, I read Animal Farm in its entirety (not a manga but its impact and meaning is so great I had to mention it), and began reading 20th Century Boys (which is a manga). I would greatly recommend 20th Century Boys. It's very non-linear though, so if you don't want to think when you read a manga, it may not be for you, but it's good.