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.
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.
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.
