Anime

I guess probably Kaiba (and it's also one of my all-time favorites).

It's odd though, because the settings and art design and such are very, very weird, but the storylines themselves - both the individual episodes and overarching plot - are actually fairly straightforward. It deals with some fairly serious themes - love, loss, loyalty, hope, betrayal, redemption and so on - and it does it very well really. It just also does it in very weird settings with very weird characters.

For that matter, FLCL qualifies in the same way. I'm not sure why I didn't think of it right off - I guess I stopped noticing how weird it is somewhere along the way, because it's just... FLCL. It is what it is. But it is very weird. In the same way as Kaiba though, behind the very weird details is a fairly straightforward story. And in a way, it's even simpler - where Kaiba deals with some relatively broad and complex societal issues, FLCL really just deals with a boy coming to terms with growing up in a bleak nowhere town, and starting to sort out how to deal with the opposite sex.

And as long as I'm here, I want to mention Ergo Proxy, which is exactly the opposite type of weird. Aside from a highly stylized science fiction setting, it's really pretty straightforward from moment to moment. It's odd, but no more odd than should be expected from the setting. But all the while, behind the current things going on, there's this background story that's bizarre to the point of near-incomprehensibility.