The Vintage Car Race
While on location in the Bahamas, Richard Lester shot some footage of the Beatles racing cars around an empty quarry. The footage was never used in "HELP!" (and was probably never meant to be part of the film), and Paul has since mentioned that the footage has been destroyed.
However, the original trailer for "HELP!" includes about 5 seconds of this lost footage. Paul is shown executing a donut, and then driving through an enormous puddle (unfortunately, it's impossible to tell who's driving the red car)! Here are some screencaps from the film trailer. It's unfortunate that this footage has been lost, because it would probably have been a lot of fun to watch!