Truth be told, I'm a little fuzzy on mine, though I distinctly remember being very proud to sound out the first pages of Green Eggs and Ham: "I AM SAM. SAM I AM." So quite possibly the first book I read on my own was that classic of the genre.
Not everyone is so predictable, though. Phantom's son LG, for example, has shown a nonfictional and artistic bent in his first solo reading choice, and it looks like his little sister is right behind him.
As for my own daughter, despite being surrounded by the cream of the crop of picture books and early readers thanks to her two librarian parents, the very first book she finished on her own, on a memorable snow day early in the winter, was this deathless title. Just goes to show you that you never know what's going to be the book that hooks a kid, and that adults' literary judgments aren't the only measuring stick.