|You ARE special, Ojou-sama!|
Naturally, people have put forth several theories on the subject of the meaning of said apple? After lurking around a bit and rereading this chapter, it finally hit me that the apple might just be a metaphor for Nagi's dreams (or perhaps Nagi and Hayate's respective dreams).
As to what those dreams are: I hesitate to say anything with committment, because it might have something to do with Hayate, it might be related to Nagi's mangaka ambition, or it might be "to be something special," which makes perfect sense as you can see in this page I uploaded.
Based on this page, we should look at the apple as an allusion to the forbidden fruit (or fruit of knowledge) from Adam and Eve, but instead of knowledge, it symbolizes unreachable dreams/wishes as far as Nagi and Hayate are concerned (especially Nagi). Notice how she says "I won't gain a thing" with the apple used as imagery.
To further back this up, the lyrics written by Kenjiro Hata himself talks about wanting something when one was small, but being unable to reach those things because they are so high until a fated meeting with someone special. Check em out at Mandoran's blog.
So that's my take on the subject. I could be wrong, but I feel pretty confident that I'm not. Feel free to discuss/dissent in the comments.