There are tons of favorites, I can say that. I won't promise to have explanations for all of these, but I'll try my best. I think I'll try to do the list in chronological order.
fairyland - The fourth Ayu song I ever heard (Because of you, is this LOVE? and evolution were the first ones) and the first one I really, truly, fell in love with. The exotic, summer-y and nostalgic sound was an instant hit for me and I also really liked what I could understand from the lyrics (my English wasn't at its best back then). I did really love Because of you too, but over the time it has dulled for me while the love for fairyland still remains.
Dearest: Random download brought Dearest to me and am I glad it did. I didn't need to read the lyrics to know this song was overflowing with emotion and I just got addicted to the song, even though it was "nothing special". A beautiful, beautiful song that can still make me tear up.
HEAVEN and
Moments: These two deepend my Ayu-fandom greatly. I got hooked on the first listen, and, what I really love about these two, I heard the emotions in these songs so clearly and it hit me like a ton of bricks, as they say. Nowadays I love HEAVEN even more, but both remain strong favorites.
NEXT LEVEL: NEXT LEVEL was actually the first Ayu-album I heard in its entirety and the titular song was my initial favorite. I seem to have a thing for these calm, breezy, summer-y songs because I loved it back then like whoa and still do. I love the chorus especially and the instrumentation is just beautiful.
Sparkle: Do I need to explain why I love Sparkle? It's weird, it's addictive, it's great. The only other early favorite from NEXT LEVEL.
winding road: I must be really weird because this was the only song that initially stuck on me when listening to MY STORY (Moments doesn't count sssssh).

I absolutely loved, and still do love, the calm arrangement, Ayu's vocals, the lovely composition, everything. It saddens me to see so much hate for this song, because it's one of my favorites.
Memorial address: I'm completely unable to recall which album came after MY STORY, but I'm pretty sure Memorial address was my next target of affection. It crept slowly into my life: at first I couldn't ever remember the song's name, just little snippets of the melody, then one day I looked up the lyrics while listening to it and BANG! tears were flowing and love begun. This is clearly one of Ayu's most emotional songs and I just love it to death. If I had to pick one song as my all-time favorite, it might just be this one.
A Song for XX: Watching the a-nation 2008 performance made me realize what an epic song this is. I was just "okay whatever I guess I like this" when listening to the original and A BEST versions, but then I got the A BALLADS version in my dirty little hands and did I love it. I still love this wonderful song to death (I think I'm starting to repeat myself).
NEVER EVER: Another song that won me over with a live performance. I had overlooked NEVER EVER before (along with most of the whole I am... -album), but seeing Ayu singing her heart out to it made me fall in love. The high-pitched verses are a bit weird (and a torture to sing grrrr damn my lack of vocal range) but the chorus is just pure win and Ayu's vocals just nail it.
Real me: I just loved this oddball of a song from the first listen. Addictive verses, nice beat and awesome arrangement won me over in a heartbeat. To be honest though, I didn't even realize she sang some English parts until reading the lyrics.
rainy day: An initial favorite from (miss)understood. Excellent arrangement, a great compositions, heartbreaking vocals and lyrics, what is there not to love?
appears: Another song that doesn't seem to get much love, I loved appears from the very beginning. Many people say she sounds bored and monotonous and lifeless in this song but I think she sounds emotional and beautiful. I really love the lyrics and the arrangement, my God, the arrangement, I get goosebumps every time I hear the intro. Such a beautiful song.
Fly high: After listening to A Song for XX and not really diggin' it, I thought LOVEppears would be a similar disappointment. Fly high changed my mind right from the beginning. The chorus is unbelievably beautiful and I just love those catchy rap parts. A great song indeed (I'm REALLY starting to sound repetitive now).
SURREAL: SURREAL was one of the many songs I completely overlooked at first listen, but later it grew on me hugely. The piano intro is divinely beautiful, and I love the unusual composition (I tend to like songs where the chorus doesn't come in right after the verse(s)), Ayu's vocals and all that jazz. One of her best songs, no doubt.
End roll: I didn't even know Ayu had a song called End roll until this summer when one summer night, I was sitting in a train with my mp3-player on shuffle. Then End roll came up and almost made me burst into tears. I regret having ever overlooked such a beautiful song.
M
Who...
fated
until that Day...
SCAR
Bold&Delicious
GREEN
untitled ~for her~
Last Links
RED LINE ~for TA~
Dolls
MOON
OKAY I THINK I'LL STOP HERE. I just couldn't write any comments on the rest of the songs, I think this post has gone to novel lenghts without those. But let it be known that I love all of those songs and more. Kudos and cookies to you if you had the energy to read through my extensive list.