Although you've stated in general what you was looking for, you haven't really made any specifics, or how patient you are willing to be. In case that doesn't make sense, I'll give you an example:
One Piece has all of what you desire (from what you stated) and more; this not only applies to you, but to a lot of other people. But because it doesn't start off with "BOOM BOOM POW POW IN YO FACE!" action and visuals, people cast it to the side. Ironically, they also exclaim that it has childish humor (evidently they didn't venture too far into it) but then state that they enjoy FairyTail... If anything, I would think that anime to be the epitome of childish humor, which gets played on way more than it should.
Well, here's a list of animes off the top of my head that I enjoyed:
Tenjou Tenge - It is quite short. It starts off pretty interesting but due to budgeting and time constraints, the ending is rushed and arguably sloppy.
Baki the Grappler - I don't quite remember how far I got, but it did hold my interest while I was watching it.
Hajime no Ippo - This is about boxing and is quite humorous, though it kind of lacks the supernaturalism you asked for.
One Piece - Love it. Great action, great story line, good use of plot elements, good use of characters and development, humorous, just about all I can ask for.
Shadow Skill - It's definitely not the best and the only reason why I liked it was because it was refreshing at the time. It is decent, it's just that the subtitles were horrible on the SUBBED version I watched, which made understanding what little story line the anime had quite difficult (but you get the gist of things).
Sword Art Online - Interesting concept that we've all fathomed at least once before this anime was created. Unfortunately, I found the quality to decrease critically after the first half (*after they left SAO*).
And then there're other animes that I have witnessed people talking about, which I did not get a chance to view or just didn't like it for my own personal reasons:
Full Metal Alchemist (+Brotherhood)
U mean neon Genesis evangelion? is it any good? I heard alot people say it was actually sorta boring but I really want to watch it soon...
Yes, master Silmarill, your answer is as follows:
It is immensely popular in Japan, that, I can say at least.
I liked it, though beware as there are now two 'versions' of it.
The original one is the anime of about 24 episodes, or 25... I don't remember, but it is somewhere around there.
Either way, they're remaking the series in the form of several new movies, new graphics and such. I like both the original series and the movies, but if you want what was(and probably still is) such a sensation throughout Japan, watch the anime, first at least.
The boring part, I guess, would be that it is a bit slow. It's a psychological type of anime as well, worth every second for me though and if you're seeking to experience an anime classic, this is what you're looking for!
All in all, it's pretty good, though I gather some have a hard time understanding it as it is a bit confusing, nothing wikipedia can't help with!