Why, the only anime (and manga) that I genuinely appreciated was Death Note, but from what I've heard and seen, it's not a particularly typical work of the genre. And what I liked about it was probably more despite any anime trappings I came across.
Tried out Neon Genesis Evangelion and although I understand what they were probably going for (at least I guess), it's not my style. Gave it up halfway through. A friend of mine forced us to watch Jin-Roh: The Wolf Brigade and Adolescence of Utena, but I was definitely not a fan. The former I at least respected somehow, though I found it kinda boring and annoying (what would Vertigo look like if you made it in Japan and Jimmy Stewart was not misguided and tortured soul, but a regular asshole?), but I hated the latter with a passion (especially the Nietzchean apologetics it tries to get into).
Wife tried to read Bleach, but I remember her not being really into it.
The most anime-like experience I genuinely enjoyed were both installments of Ori I played in these past few months. And sure the developers themselves admit they were influenced a lot by Hayao Miyazaki (whom I've yet to discover), so I think it's fair to put it here.
I would like to at least try Miyazaki, Fullmetal Alchemist, Ghost in the Shell, Code Geass, Attack on Titan and especially Haibane Renmei and Cowboy Bebop ... but I honestly don't know when, 'cause like I said, I haven't really liked the stuff I came across so far.
Also what doesn't help is the particular friend mentioned above - unfortunately he's a huge anime fan and quite a typical one at that. And having obscure anime references / analyses in pretty much every e-mail he sends tends to wear you off a lot.