I think TO BE, Who..., untitled ~for her~, teddy bear, and A Song for XX move me the most. Ayu does have the knack of conveying emotion through her music that I seldom feel from other artists. This is the reason why she exploded in popularity in the first place (apart from her image) because people who listened to her music felt like she could understand them.
Songs like A Song for XX and SEASONS resonate in people's hearts!
This is truly her talent. Whether the emotions are truly her own, or if she's emulating someone else's emotions into her own music, she does it so well that it comes alive. I think she takes advantage of this talent really well.