I've been trying to get my sister into anime, and the first series I got her to watch all the way through was Hikaru no Go, and she said it ended perfectly (minus the special episodes, which had something of a tacked-on-after feel). It's more in the coming-of-age genre, so not a tragic ending like OP likes, but Hikaru does come-of-age quite nicely.
Another couple w/ awesome endings, more in-line w/ the OP tragic endings likes, and Gungrave & Berserk. Gungrave is all Mafia/Gangster/Zombies w/ themes of loyalty & friendship, and a kinda bittersweet tragic ending. Berserk is unusual in that the anime doesn't cover near as much as the manga, but while most of the animes like that suck, the Berserk anime is better for it IMO. The manga is still being written today, and has kinda the same we're-dragging-this-on-and-on-because-we're-popular-and-we-can feel that Bleach and Naruto have. Although IMO the manga is much better than Bleach or Naruto, I would probably recommend just watching the anime & leaving it at that - although the ending does make you want to read the manga.