Addition: a few anime soundtracks, just to make a point that not all anime music is necessarily
cheesy cutesy crap that even I (after all these years) still can't stand. I don't mean even to try to be exhaustive; the selection is personal.
https://www.youtube.com/watch?v=_Eb4ONZw01YMahou Shoujo Madoka Magica ED "Magia"
The animation on the ED is rather lackluster, but I just have to add at least one piece from Yuki Kajiura.
https://www.youtube.com/watch?v=OM8ODecbuhASayonara Zetsubou-sensei: 1st OP "Hito toshite Jiku ga Bureteiru"
Seeing this opening scene surprised the crap out of everyone when it first aired. No, this is not a r-rated BDSM show.
https://www.youtube.com/watch?v=lX-yENHkeMoK-on! ED "Don't say 'Lazy'"
The voice actresses from the show actually formed a band, producing music as the fictional band within the show. Wrap your head around that..
https://www.youtube.com/watch?v=whK7sZacKzAK-on! Soundtrack "Sweet Bitter Beauty Song"
Released on the same single with "Don't say 'Lazy'" above. I actually wish this song played a bigger role in the anime.
https://www.youtube.com/watch?v=WWB01IuMvzASuzumiya Haruhi no Yuutsu Soundtrack "God Knows" and "Lost My Music"
Actual anime footage, full versions easily found on YouTube. If you like the song, and if you're bored, take a look at the Russian dub of the scene; in many ways it's superior.
https://www.youtube.com/watch?v=AkiNUNC4ZYUHigurashi on naku koro ni OP, "Higurashi no naku koro ni"
I felt like I was going insane as I repeated the track on the ipod for hours on end when the anime was airing. I shouldn't have done it.
https://www.youtube.com/watch?v=soGzGCxW27cNoir Soundtrack "Salva Nos"
Other than a couple of Latin mistakes such as using "Dominus" instead of proper vocative "Domine", this one used to be my top for a long time. Actually, I think it still is.