The swimmer who pops up twice asking for the White Cliffs of Dover, once in the Alps, and once in the Bahamas, is Mal Evans, road manager for The Beatles.
One of this movie's original taglines was, "Please do not reveal the beginning of this movie to your friends (they'd never believe it, anyway)". This is a spoof of the tagline from Sir Alfred Hitchcock's Psychose (1960), which implored its audience, "Please do not reveal the ending of this movie to your friends (it's the only one we have)".
While The Beatles are attempting to solve Ringo Starr's ring problem at the jeweler's, George Harrison can be seen quietly shoplifting various jewels and slipping them into his overcoat pockets.
By Paul McCartney's and Ringo Starr's own admission, they were so stoned on pot the day they shot the scene where Dr. Foot (Victor Spinetti) and Algernon (Roy Kinnear) tried to blow them up in the Austrian Alps that when George Harrison screamed his line "It's an fiendish thingy! Run Ringo!" both Ringo and Paul ran over the next hill.
According to Sir Paul McCartney, the script was designed around their requests that the story involve them going to places like the Austrian Alps and the Bahamas because they had never been there before. (The Bahamas were also chosen because they were British territory, and allowed them a tax shelter.)