Galileo's writings describe two experiments involving ships. These are summarized in the Wikipedia article Galileo's ship. (A lot of the text in the article is mine, and if there are things I'm misunderstanding, then they probably appear in the article as well.) In one of these experiments, a cannonball is dropped from the mast of a moving ship, and is observed to land at the base of the mast rather than behind it. In his letter to Ingoli, he insists strenuously both on the fact that he really carried out the experiment and on the outcome.
This experiment is basically identical with one shown in the classic PSSC film Frames of Reference. For 20 years, I've been showing students this film, and accompanying it with a description of Galileo's version, and an anecdote I heard somewhere about a bet. The story is that Galileo got in a bet with two noblemen about the outcome of the experiment, but after the noblemen agreed to the bet, Galileo was unable to collect his money because they didn't agree with him that an experiment was the proper way to settle who was right. As I've learned more about the history in recent years, I've failed to come up with any information to support the anecdote about the bet. Is it totally bogus?
As circumstantial evidence against it, secondary sources such as Drake devote considerable energy to discussing the evidence that the experiment was real rather than a thought experiment, and, e.g., Drake never mentions a bet. It seems that if there were historical sources for the story of the bet, then a specialist like Drake would have considered them important evidence on this point.
Many other people from that era appear to have discussed the experiment or claimed to have done it. Maybe the story about the bet is really about one of them, not Galileo?