Alice and Bob are playing a harmless game like foosball or table tennis. Alice is the classic "I'm not allowed to lose!" bad winner, who tends to have victory dances. Surprisingly, Bob beats her.
Does Alice simply say something along the lines of "Good game"? No, most often, she says, "Best two out of three!" And she loses again. "Best three out of five!" Another loss. "Best four out of seven!" And so on along the formula "Best x+1 out of (2x+1)" where x is how many times Alice has lost. Occasionally, it gets to the point of "Best 400 out of 799!" Often, the numbers used make no sense (e.g., "Best four out of nine").
A variation of this comes if there's money riding on the game, or another inconsequential wager. Then, Alice's apt to offer "Double or nothing?" to get her shot, and is likely to keep going "double or nothing" until she bets something she really shouldn't have.
Ranma 1/2: Akane suddenly gets much stronger because of some special noodles. To test whether she really had become insanely strong, Ranma challenges her to arm-wrestle him. He loses, but promptly challenges her to another round. This continues until Akane gets bored of it and Ranma is battered and beaten. When she changes back he insists on continually extending it even though he is winning to celebrate.
In Minami-ke, Kana starts one of these with Fujioka to see who can get higher test scores. He ends up beating her on every test, despite the fact that she changes her scores to make them look higher.
Touma: You say duel, but you've lost every time, haven't you? Mikoto: Shut up! You haven't actually landed a blow on me, so I haven't lost. Touma: When's this gonna end, then? Mikoto: Obviously, when I win.
In Heaven's Lost Property, Tomoki is humiliated after scoring the lowest in a trivia game show. In an attempt to prove he isn't dumb, he challenges The Ditz Astraea to a game of Go, but she easily wins. He angrily challenges her to more games, degenerating from Shogi to foosball, video games, baseball, etc. with Nymph lampshading that the later games don't prove intelligence. Astraea wins 26-25.
In Azrael Annual #2, a Legends of the Dead Earth Elseworld, Two-Face does this to himself. When Azrael meets him, he's standing at a crossroads tossing a coin to decide if he is good or evil. We're not told what answer he wants, but Azrael leaves him going "Two out of three ... no, three out of five ... no..."
In Bill & Ted's Bogus Journey, the title characters attempt to win their freedom in a game of Chess with Death in a sequence reminiscent of The Seventh Seal. Only instead of chess, the boys play Twister, Battleship, and a variety of other contemporary games, which Death keeps losing. (Of course, Death doesn't want to admit he's losing to these airheads so he keeps altering the conditions of the challenge, making it "best 2 out of 3", "best 3 out of 5" and so on...)
This is actually used as a plot point in the original script, which was used for the comic book adaptation. At the Battle of the Bands, Bill and Ted are repeatedly killed by their robotic Evil Counterparts, and they make Death bring them back each time, citing the number of games at which they beat him.
The novel drops the number of resurrections to 2, saying he (Death) owed them one life per game (so 4)
Appears in the Daredevil movie. A darts player in a pub asks "Best of three?" after being beaten by Bullseye, who scored four bullseyes in a row.
In Toy Story, with Mr. Potato-Head, losing at Strip Battleship with Hamm, says "How about three out of five?" after losing his nose.
An inversion in Enderís Game: Ender challenges an older student to a game, but immediately states that he is going to win two out of three. The student wins the first match and tries to claim victory, but Ender convinces him that there's no pride in winning once against someone who've never played that game before and proceeds to win next two matches.
A variation appears in an episode of Frasier in which Frasier learns that Martin is a better chess-player than him; no one actually says 'Best out of (however many)', but Frasier won't stop forcing Martin to play chess until he wins a game.
An episode of the original serial The Addams Family has Gomez lose to Morticia in every possible match (from fencing to chess), reaching ludicrous numbers.
The Double-or-Nothing variant pops up in an episode of Friends, ultimately culminating in Monica and Rachel losing their apartment (and mattresses) to Chandler and Joey.
Also in Friends Monica and Phoebe's future-husband Mike are driven to play ping-pong out of boredom. It is revealed that they are both as competitive as each other, and the game gets more and more frenzied as Monica's hair gets bigger and bigger.
Serious and odd variant in Farscape, where the doubled Crichton(s) play rock-paper-scissors over and over again, in complete silence, hoping to not tie with each other and prove they're not identical.
Played with on the old Disney show The Jersey. Nick and Cole competed in a series of strange athletic challenges of their own device, with wins trading back and forth. Every time they had a clear winner, the clear loser would raise the numbers again, then (presumably) win the next two events.
In Scrubs, the janitor makes an unlikely flick of cotton balls across the desk. Kelso bets he can't repeat the trick, which the janitor accepts. After failing, he keeps betting double or nothing, to the point that he owed Kelso hundreds of dollars, until the end of the episode where he makes it. "We're even, goodnight sir."
Seinfeld had the double-or-nothing variation when Elaine kept betting Jerry that she was broken up for good with Puddy. Several rebounds later, she gives up, stops betting, and Puddy breaks up with her.
Happens in Corner Gas when Emma tries to reclaim some money her husband gave to Hank. In this case it's deliberate, since she knows Hank is better at pool than her. The goal is to keep playing until Hank simply hands over the money so he can leave.
Rugrats did this trope straight-up, with Angelica challenging Susie to "two out of five", "two out of nine", and "two out of fourteen".
Happened in The Flintstones with a coin toss over a winning cruise ticket (only good for two people), except it kept being extended by the *winner* who felt sorry for the losing couple. Eventually, the two ladies ended up going together.
The same variant as in Frasier occurs in the Chowder episode "The Cinimini Monster": Chowder and Mung are held captive by the titular monster, and are forced to play board games and win to be freed. However, whenever one of them gets ahead of the monster, he immediately throws the game off the table and switches to a new one. When Truffles is called in to beat the monster (which she does) he locks up the house and swallows the key.
Hey Arnold! once featured the title character challenging a man living in the city sewers to a game of chess with the plot's MacGuffin serving as reward. Arnold wins every game with the sewer man eventually demanding "Best 8 out of 15" games.
In the My Little Pony: Friendship Is Magic episode "Dragonshy," Rarity and Pinkie Pie play tic-tac-toe while waiting for Applejack and Fluttershy, with Pinkie Pie winning 35 straight games. Rarity grumbles about this, then brightly proposes, "Best of seventy-one?"
Occurs at the end of the Johnny Test episode "The Enchanted Land of Johnnia" after Johnny and Dukey find out they were actually in a virtual reality game. Johnny's sisters win and Johnny challenges them to rematches twice with the usual "best two out of three" and "best three out of five"... but after he and Dukey win the first time, the girls challenge them to "best four out of seven." They could've just won the next game to beat Johnny and Dukey's last challenge instead of making it harder for themselves.
In one episode of Danny Phantom, Jack and Maddie kept playing games. Jack, who kept losing, suggested "best 2 out of 3", "best 3 out of 5", and so forth.