Here are some recommendations for further reading, in books by Hofstadter or by others.
If you want more recreational math:
Hofstadter, Douglas. Metamagical Themas. New York, NY: Basic Books, 1996. ISBN: 9780465045662.
A hodgepodge of all things Hofstadter, collected mostly from his Scientific American columns. Includes the group theory of Rubik's Cubes, an attempt to explain LISP to a general audience, a theory of fonts, and the beginning of his "letter spirit" analogy puzzles.
Gardner, Martin. The Last Recreations. New York, NY: Springer-Verlag, 2007. ISBN: 9780387258270.
Just about anything by John Conway, but try Conway, John Horton, and Richard K. Guy. The Book of Numbers. New York, NY: Copernicus, 1996.
If you want more about languages:
Hofstadter, Douglas R., and Clément Marot. Le Ton Beau De Marot: In Praise of the Music of Language. New York, NY: Basic Books, 1998. ISBN: 9780465086450.
This book is about translation and the endless creativity involved in using natural languages. It's a lot of fun, even though Hofstadter's poetry sucks.
Deutscher, Guy. The Unfolding of Language: An Evolutionary Tour of Mankind's Greatest Invention. New York, NY: Holt Paperbacks, 2006. ISBN: 9780805080124.
The clearest, most interesting overview I have ever seen of how and why languages change, all done without unnecessarily assuming untested theories like Universal Grammar.
If you want more philosophy:
Hofstadter, Douglas R., and Daniel Clement Dennett. The Mind's I: Fantasies and Reflections on Self and Soul. Toronto, Canada: Bantam Books, 1985. ISBN: 9780553345841.
A whirlwind overview of modern philosophy built out of many essays and short stories, with Hofstadter's commentary added after each chapter.
Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 2004. ISBN: 9780691121277.
Okay, I haven't read this, only skimmed it in a bookstore so far, but Rudy Rucker seems to have a freewheeling writing style reminiscent of Hofstadter.
If you want more AI:
Hofstadter, Douglas R. Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought. New York, NY: Basic Books, 1995. ISBN: 9780465051540.
This is the most "academic" of Hofstadter's books, and you probably won't get anything out of it unless you are specifically interested in analogy-based AI.
Minsky, Marvin Lee. The Society of Mind. New York, NY: Simon and Schuster, 1988. ISBN: 9780671657130.
Minsky presents his ideas on the structure of the mind in a unique format - one chapter per page, with copious references between chapters.
Kurzweil, Ray. The Age of Spiritual Machines: When Computers exceed Human Intelligence. New York, NY: Penguin, 2000. ISBN: 9780140282023.
Dated by its 90's optimism, but still an interesting read (and also built around dialogues).
I don't recommend reading Hofstadter's new book after GEB, because it duplicates too much of its content.