Thanks to some investigative journalism over at Equestria Daily, the third song that the Twilight Sparkle Animated Storyteller sings has been figured out! It plays the intro theme, Winter Wrap-Up, and a third, unknown song. If just want to hear the third song, skip to about 1:55 in;
An inquisitive fan on Facebook asked Daniel Ingram, the composer for Friendship in Magic, what the third song was, and he was kind enough to answer (click the picture to see a bigger version).
Makes me wonder how many songs were recorded that we didn't hear!