Every Otaku has theirs and I'm no different and after watching about two decades worth of anime, I'm just as qualified (which is to say everyone is who can be bothered to have an opinion) as anyone else to have a list of favorites.
I'll try and do my best at embedding some links to the music but sometimes, me and embedding don't get along so much.....Artists given credit when I have the information to share.....
from the bottom up:
#10) Inuyasha - Season 1 - It has that uplifting feel to it that makes you feel as if you're about to embark on a grand adventure and as such, gets you in the mood for what is to follow.
Change The World/V6
#9) Azumanga Daioh - Let's face it, this theme music is an ear worm that can stay with you for days and give you problems pronouncing the word Cake for weeks at a time.....
#8) FLCL - Perhaps this is too low, I dunno, I just know that I have to have this on the list somewhere and here's as good a placeholder as any at the moment.
Ride on Shooting Star/ The Pillows
#7) Love Hina - My first and still favorite Harem comedy and this manic opening just continues to make me smile and never gets old as far as I'm concerned.
#6) His and Her Circumstances/Kare Kano
Much like the entry at #1, it's not just the opening sone, but all of the other incidentals and closing music that forced me to go out and buy a series soundtrack... that doesn't happen with every series. The voice is so painfully beautiful that I tear up every time I hear it and I guess that just makes me waaaay too sentimental. Definately a ymmv selection I suppose.
#5) Sakamichi no Apollon - more incredible music from Ms. Kanno
#4) Mysterious Girlfriend X - Again, this one is more for the entire show rather than simply the opening itself, because the music is incredible throughout with setting mood and conveying unspoken emotion. The Dream sequence segments themselves are inspiring.
#3) Beck aka Mongolian Chop Squad - dunno how this is left off of so many folks lists, but considering this anime is just as much about music as it is about self discovery, I have a hard time wondering how this isn't on EVERY list.
Hit in America/Beat Crusaders
#2) Bleach - Season 10 - Dig the song, stay for the anime
Scandal/ Shoujo S
#1) Cowboy BeBop - - C'Mon, you knew it was coming... the cultural rarity to include multiple genre's and still allow the music to aid in the story telling effortlessly, seamlessly. Scary when you dare to try and separate the work from the music and you realize that to do so would be a disservice to both.
Tank/The Seat Belts
Kimi No Todoke - http://www.youtube.com/watch?v=8AzEFia1UEY&feature=related
Lovely Complex - http://www.youtube.com/watch?v=fXeHfjpt7WE
Ghost in the Shell - http://www.youtube.com/watch?v=NiiFtS22yyM
Gunslinger Girl - http://www.youtube.com/watch?v=2NSvaT6h52s
Higurashi No naku Koro Ni - http://www.youtube.com/watch?v=rKV195eXg7I&feature=related
Durarara - http://www.youtube.com/watch?v=_O9hRcIAza0&feature=related
As you can tell, I'm kind of all over the map here, but hell, I just go with what I like and what has made an impression with me, as always, your mileage may vary.....