Galileo's pupil Viviani said that Galileo dropped unequal weights from the Leaning Tower of Pisa and observed them to take equal times to hit the ground. Galileo's own writings do not describe such an experiment, but it is clear that he did quantitative experiments with balls rolling down inclined planes, which would make it natural to extrapolate to the same result for falling objects. (I suppose it's not a matter of pure extrapolation, since the acceleration of a sphere rolling without slipping is strictly less than $g\sin\theta$.) Wikipedia says that most historians of science think there was never any more than a thought experiment (e.g., Koyre seems to have thought this), but that Drake disagrees.
What would be the arguments for and against the actual existence of the experiment? What did Viviani claim Galileo actually did? (Apparently in some versions of the story, Galileo dropped his objects near the heads of professors emerging from their lectures!)
Is the only argument about whether the experiment took place at the Leaning Tower, or is there doubt about whether it happened at all? Galileo does describe such an experiment, in a line spoken by Simplicio:
But I, Simplicio, who have made the test, can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits.
There seems to be room for doubt, since it's fictionalized as a dialog, and the words are spoken by Simplicio rather than by Salviati, who is Galileo's mouthpiece.
As a side issue, there appears to have been some controversy about what Aristotle had really claimed about this. Apparently there is a case to be made that he was only talking about terminal velocities. For translations of the relevant passages from Aristotle see this link.