My favorite part of David Wallechinsky's quadrennially indispensable The Complete Book of the Olympics are the discontinued sports and events, such as Motor Boating (1908 London games), Tug of War (1900-1920), Croquet (1900 Paris) and Jeu de Paume (or real tennis, which was won in Paris in 1900 by robber barron Jay Gould's son Jay Jr.).
The 1900 Paris Olympics went on intermittently over a five month stretch during the World's Fair to such little local interest that some athletes didn't realize they were competing in the Olympics. An American socialite girl won the gold medal in Ladies' Golf in Paris in 1900, shooting 47 for nine holes, but died in 1955 before learning that the little tournament had been an Olympic event and she was an Olympic Gold Medal winner.
Cricket's entire Olympic history consists of a single match in Paris in 1900 between Great Britain and France (a team recruited from the British Embassy in Paris). A French magazine explained, "Cricket is ... a sport which appears monotonous and without color to the uninitiated." A British observer reported, "We found the French temperament is too excitable to enjoy the game and no Frenchman can be persuaded to play more than once."
Discontinued track and field events include the Stone Throw (1906 "intercalated" Athens games), Javelin (Both Hands) (1912 Stockholm -- I don't know if this means they summed up your best throws with both your left and right hands, or if you were supposed to heave it two handed ... or did they throw javelins with each hand simultaneously?), and Triathlon (1904 St. Louis -- a combination of track and field and gymnastics).
American Raymond Ewry is the Carl Lewis of the Alternative History Olympics, winning ten track and field gold medals from 1900 to 1908 (including two in the now-forgotten 1906 Athens Olympics) in no-longer-existent events: Standing High Jump, Standing Long Jump, and Standing Triple Jump.
Gymnastics once had Rope Climbing and Club Swinging, Shooting had Dueling Pistol and (by popular demand among commenters) Live Pigeon Shooting, and Swimming featured the 200 Meter Obstacle Race.
My favorite of all the NLE swimming events is the Plunge for Distance, won by William Dickey of the USA at the 1904 Olympics by diving in the pool and not taking a breath and not taking a stroke until he had glided 62.5 feet. I've always had a feeling that if this event were both still around through sheer bureaucratic inertia and yet still unpopular and not very competitive, that I could have won an Olympic Gold Medal in it. Granted, I've never actually measured how far I could plunge, but I've always had a knack for floating face down.
I've sometimes dreamed that I had won an Olympic Gold Medal at some point in the past. When challenged in my dream about this unlikely accomplishment, I've tried to parry, "Yes, it was once a part of my life, but I choose not to talk about it much these days." When grilled on exactly which event I could ever have won, I usually come up eventually with "Plunge for Distance at the 1984 LA Games -- you probably don't remember it because it wasn't on TV." My dream interlocutors go away, grumbling, but unable to prove me wrong.