My 1st favorite anime intro song would be Tank by the Yoko Kanno and the Seatbelts for the anime Cowboy Bebop. Even the ending song was great. The Old Folk Blues.
YouTube - cowboy bebop opening
YouTube - Cowboy Bebop Ending
My 2nd favorite would have to be from DeathNote. Maximum of Hormone
Whats Up People!.
YouTube - Whats Up People-Maximum Hormone
My 3rd favorite would be Kiri by Monoral for the anime Ergo Proxy and the ending of Ergo Proxy Paranoid Android by Radiohead.
YouTube - MONORAL - Kiri (Ergo Proxy OP)
YouTube - ergo proxy ending title!
Now the reason I like the Cowboy Bebop intro and ending songs is because I could relate to them. Most of the time I feel old and out of place. Im old school in other words and these songs being of jazz and blues (once popular music genres but now are an old form of music).
Whats Up People is just one of those songs that introduced me to an entirely unique band. And I have a habit of collecting unique things. But the reason I truly like the song is its different sound. It has a bit of a System of a Down feel to it but a more heavy metal punk feel to it.
Monoral is one of those bands I just want more of their music. They have that very relaxed sound to it including Radiohead. But I think what makes me even like the songs more is the anime that they are apart of. Ergo Proxy was so philosophical and these songs helped begin and end each philosophical episode with their deep meanings.