Absolute Anime

Bleach-Asterisk, Thank You, most of the other stuff.
Trigun-HT(and all the TRIGUN music rocks)
Bobobo-bo-bo-bobo-Wild Challenger
FMA-Ready Steady Go
Club to Death Angel Dokoro Chan-{title same as show)
Dragon Half-End theme (Priceless. you must hear it)

Full Metal Alchemist- Ready, Set, Go!
Inuyasha- Itanzuara No Kiss
Yu Yu Hakusho- The Sun will Shine again

Ready Steady Go!-FMA
Reckless Fire-S-cry-ed
RK ending theme-RK
All Inuyasha opening and ending themes
Bokutachi No Yukue-GSD
Invoke-GS
Ending #1 Theme-GSD
Just Communiaction-Gundam Wing
The Cruel Angel's Thesis-Neon Genesis Evangleion
Tobira no Mukou he-FMA
Battlecry-Samurai Champloo
Gundam Seed Theme-Gundam Seed
Ignited-GSD

i love the songs:

ready steady go from FMA

change the world from inuyasha

soramimi cake from azumanga diaoh

"Go Where No One's Gone Before" L/R. Peace out.

Ready Steady Go-FMA
Reckless Fire- s-Cry-ed
Bokutachi No Yukue-GSD
Invoke-GS
Ignited-GSD
Pride-GSD
Rewrite-FMA
Ichirin no Hana- Bleach
Asterisk-Bleach

I like the real folk blues from cowboy bebop

if you mean opening, my all time fave OP would be "Cruel Angel's Thesis" from evangelion. but if it's the overall favorite anime song mine would be "Nowhere" from Madlax, its so addictive.

Enigma wrote:Nearly every music from Nagareboshi Gin is great, especially the intro theme.

Pegasus Fantasy from Saint Seiya is also very good.

Getta Robo´s theme ain´t bad either.

Nice to see someone else that actually knows what Getter Robo is

Anyways to add to my original list

Chou-denshi Bioman (It's live action, but it IS Japanese, so I guess it counts)
Biomic Soldier (Yes, it is spelled Biomic...)
*both of these songs are from Bioman*

Lonely Soldier Boy- Genesis Climber Mospeada
Gaiking- (By Psychic Lover)- Gaiking: Legend of Daiku Maryu
Heat- Shin Getter Robo
Getter Theme- Getter Robo (How on earth could I have forgotten this song?)
Lonely Way- SPT Layzner
Matamorphoze- (By Gackt)- Zeta Gundam: A New Translation Movie 1- Heir to the Stars
Ace Attacker- SRW@
Vanishing Trooper- SRW@

Basically all the theme's from:

Bleach
Black Cat
Wolf's Rain

Haven't seen so many anime's yet.

inu yasha - come (ending theme)
naruto - ryuusei
pichi pichi pitch - mother symphony, legend of mermaids

fma
cardcaptors
pokemon
yugioh
sailor moon
saint tail
and many more

Life - Bleach
Life is like a Boat - Bleach
Lily - Elfen Lied
Minamikaze - Full Metal Panic? The Second Raid!
Mouichido Kimi ni Aitai - Full Metal Panic? The Second Raid!
Eternal Snow - Full Moon wo Sagashite
beginning song of Ichigo Mashimaro
Deep Forest - Inuyasha
Dearest - Inuyasha
Every Heart - Inuyasha
My Will - Inuyasha
Love Tropicana - Jungle wa itsumo Hale nochi Guu
Kirei na Kanjou - Noir
Lullaby - Now and Then, Here and There

Of course, I like most anime songs, but these are just a select few. ^__^

My favorites are all the songs of Fruits Basket.

Reckless Fire-s-CRY-ed http://www.youtube.com/watch?v=Yxdm4bAeiKI

cool i will find some