One that isn't too disgusting or too weird to comprehend? It's dark, it's thought-provoking and it shows the not-so-nice side of current (that is, not futuristic) Japanese society... Paranoia Agent. It doesn't have too many episodes, but it gives you so much that children's or young adult anime would never give you, like new perspectives on how technology may be keeping us apart, how unreliable people can be, how easy it is to hide oneself in fantasy... it's quite good so far, but to be completely honest I haven't finished it.
I still don't quite understand the mythology, but Serial Experiments Lain has been mentioned, with good reason. Again, not so dark that I lost interest in the hopeless mess of it all, but dark enough to make you shudder and consider things you otherwise wouldn't have wanted to. This one is a bit less real-life at the beginning, but it doesn't become too supernatural at the very end like Paranoia Agent kind of does. The story loops back on itself so much I still have some puzzling out to do, but at least I finished watching it.
In movies you have a much bigger adult selection. I (fearfully) recommend Akira, but that one even made me feel ill, so be prepared for body horror. There's not much you need to think about like the other two I mentioned, this one has just about all the mystery solved by the end. Good for if you have less time to spare for watching and figure out the plot. It is, as you requested, most certainly not for children.
