According to his biography, "I'm Chevy Chase and You're Not", Chevy Chase filmed a version of the joke as well. However, at the last second he decided not to sign the release form as he was afraid of what his daughters would think of the joke's content.
This joke has been used by comedians for years as a mental stretching exercise. They would tell the joke to each other as a warm up for their acts. While the setup and punchline remain the same throughout, the middle section, describing the actual family act, is always varied to get the juices going by throwing in the most they can and keeping it funny. There are stories of parties where performers will jam and keep it going for almost an hour.
On his radio show, co-director Penn Jillette said that Rodney Dangerfield and Buddy Hackett were both invited to appear in the film and were supportive of the film, but declined due to their failing health (they would both die before the film premiered). Also, the filmmakers intended to have a private screening for Johnny Carson at his home, but he died only days after the premiere at Sundance. They then decided to dedicate the film to him.
Terry Gilliam's interview was cut apparently because of an unforeseen sound error during the taping. He appears in the DVD extras, though, with voice-over from director Paul Provenza who first talks about their chat, and then adds he learned that "a director should always wear headphones".
Promos for this film promised a film with no violence or nudity but tons of bad language. However, Penn Jillette had to add blurring to the private parts of a "friend" sharing a jacuzzi with one of the comedians in order to fulfill this promise.
According to T. Sean Shannon, he got into trouble behind the scenes at Saturday Night Live (1975) for his use of the N-word in the film. Shannon also said that he had been berated by African-Americans when recognized in public.