Story wise Evangelion, it's been mentioned, but I haven't ever seen much weird anime. FLCL as a close second, but that's more insane than anything. I'm sure if I watched Hitalia or Panty X Stocking I'd have my mind blown.
Back to Evangelion It's weird not just because of the obvious stuff, but look at the story, (series wise only) we have an unlikeable whiny protagonist, like three sets of antagonists all working against him, and each other. So many odd things, plot points out of nowhere, the lance of longius, Touji in EVA 03, Kawaroru, all just showing up, doing what they need to, and abruptly leaving.
Not to mention the batshit insanse conclusion, complete dynamic shift out of nowhere. We go from fairly campy monster of the week robot war with occasional nightmare/depression fuel, to exostentialist philosiphy time.
And then that resolution, Shinji doesn't have to fight for anything, he doesn't really overcome anything, he just learns to get some self confidence. Not to mention how now all the bad times are gone, he can live on that weird planet thing and fondle Rei all Oedipus complex.
Oh and that one high school alternate reality, did not see that coming, slice of life from out of nowhere. Sorry for the wall of text.