(See the Related links for a really interesting video on the history of the logo: Tiki Barber interviews various members of the Green Pay Packers themselves.)
I am pulling just a few of the comments from the discussion page (which many may fail to access and read), as it seems that many contributors have a conflict with the original response and therefore should not stand on this page alone. (Personally, I would note that the term "Greatness" would seem to be the wrong variant/tense for what it is suggested it was for: If there is any truth to this story, one would think the term being promoted would have been "Great" or "Greatest".
-- The oval "G" logo which stands for the "G" in Green Bay was created in 1961 by Packers equipment manager Gerald (not George) "Dad" Braisher. Tiki Barber falsely reported it to stand for "greatness" without a reliable source to back up his claims; along with calling him George instead of Gerald. Other reputable media outlets then published similar stories using Barber's false claim as a source. The Packers' Assistant Director of PR and Corporate Communications had the following to say: "There's nothing in our history that suggests there's any truth to this."
-- After George retired in an interview with a local newspaper he was quoted as saying he ordered it as a joke. He wanted his initial on it and thought when he submitted the order to the owner for approval the owner surprised him by approving it. About a day later it dawned on him "G" could also stand for "Green Bay".