[snackbroken]

Seconding Fullmetal Alchemist. I hear the remake (Fullmetal Alchemist: Brotherhood) is usually regarded as the better version. More suggestions: Neon Genesis Evangelion, Death Note, Sousou no Frieren, Cowboy Bebop, Nichijou, Tengen Toppa Gurren Lagann, Bakemonogatari. There's also quite a few good movies, anything by Studio Ghibli is great, and so are Akira, Perfect Blue, and Ghost in the Shell.

[ThrowawayR2]

Some of those those aren't really going to appeal people unfamiliar with the conventions of the genre and some of the big personalities, e.g. Evangelion is a deconstruction of the once popular giant robot genre and Hideaki Anno's personal couch trip rolled into one.

[trenchpilgrim]

Brotherhood follows the plot of the source material comic, which is regarded as having a better ending. The original series aired concurrently with the comic and had to diverge when it passed where the ongoing comic ran out of chapters.