The first one that jumps to my mind is Forgiveness from Power of Music.

I prefer a lot of the Power of Music live versions over the original versions, I think Power of Music is the highlight of her career actually

I also prefer the live versions of Memorial Address and count down from the Rock'n'Roll Circus tour

Nothing can beat 'M (unplugged version)' from AT02

this this this this
she has never nailed a song as much as that performance.
the rain, the voice, the attitude, the everything.
I downloaded it in 1080 and watched it about a million times. almost felt like i was there

Whatever - CDL 2004-2005. Love it when the guitar etc kick in!

mi|kshake~

SURREAL ~ COUNTDOWN LIVE 2005-2006

evolution ~ COUNTDOWN LIVE 2009-2010 A 〜Future Classics〜

M ～POWER of MUSIC～ 2011 A LIMITED EDITION

and then ~ concert tour 2000 A

WHATEVER ~ Countdown Live 2004-2005

PoM version of M is the best!

November (CDL 11-12)
Hanabi (a-nation 05)
forgiveness (Power of Music)
Bold&Delicious (CDL 05-06)
flower garden (A Museum 2003)
surreal (Dome Tour 2001)

Evolution from Asia Tour 2007. I love Evolution live in Ayu's later concerts better than the Studio recording, but I love the Asia Tour 2007 version the best.

Yeah!
I have it on ringtone :-)

Am I the only one here who loves RAINBOW from COUNTDOWN LIVE 2007-2008? :P

Uhm, I'd say..
～POWER of MUSIC～ in general
evolution (ASIA TOUR 2007)
A Song for ×× (a-nation '08)
SEVEN DAYS WAR (RNCR ～7days Special～)
RAINBOW (CDL 2011-2012)
GREEN & Will (PCDL 2008-2009)
and many others

:yes