Played with in "The King of Omashu", where the extremely aged and hunchbacked King Bumi gives Aang a choice between two equally fearsome looking opponents. Feeling clever, Aang picks Bumi himself, who turns out be one of the most powerful Earthbenders in the world and promptly kicks Aang around the arena like a football. The whole point of this and the other exercises were trying to think outside-the-box, and Bumi made the third option so obvious by saying "choose your opponent" and standing right in front of him that taking the third wasn't really that creative.
King Bumi: Heh heh, wrong choice!
When Aang returns to Omashu while it is being controlled by the Fire Nation, he sought to rescue an imprisoned Bumi. Surprisingly, the old king can still earthbend, but has decided not to yet. Aang knows of the way of positive jing, fighting, and the way of negative jing, or retreating. Bumi reveals that there is a third "neutral" jing, which involves doing nothing and waiting for the right moment. This third jing would be key to Aang's mastering of earthbending.
This same principle pays off for Bumi when the eclipse arrives, rendering firebending useless and allowing him to single-handedly take back the city.
This trope crops up again in the series finale, where Aang is forced to decide whether to let Ozai live and carry out his genocide on the Earth Kingdom, or kill him outright. Aang manages to get around this by learning how to Spiritbend by getting Touched by Vorlons and permanently disabling Ozai's ability fo firebend.
Done for humor in the Batman: The Animated Series episode "Almost Got 'Im". Harley Quinn captures Catwoman and ties her to a conveyor belt heading for a massive meat grinder. Batman arrives, and catches Harley, who then taunts that he can either bring her in, or rescue Catwoman, but not both. Batman then... nonchalantly reaches over to the circuit breaker and shuts off the power to the grinder, to which Harley responds, "heh heh... Good call—Help!"
Defied in The Batman, where D.A.V.E., a robot programmed as "Gotham's Ultimate Criminal Mastermind", forces Batman to choose between Alfred's life and his secret identity. Batman tries to free Alfred through different means, only for D.A.V.E. to slam him against the wall of the Batcave, shouting that he knew he would try to take a third, more favorable option.
In one episode of South Park, Towelie was faced with either preventing the boys and their new game system from falling into a death trap, or getting high from a joint the evil towel was taunting him with. Towelie's response? "I choose.... BOTH!"
In one episode of Family Guy, Peter and his father-in-law sell Meg some marijuana, creating an implicit choice between the money and the pot, so Mr. Pewterschmidt hits Meg over the head and declares "Now we have the pot and the money!"
In the X-Men animated series, Bishop goes back in time to stop Apocalypse from causing a global plague. But in Cable's time (further into the future), Cable realizes that if Bishop saves the present, it would doom his future. The plague would allow humanity to develop antibodies that would help the people in Cable's time survive further plagues. So, if Cable wants to save his people, he has to help Apocalypse win. His third option? Expose Wolverine to the virus so his Healing Factor would create antibodies to counter the virus, thus giving it a cure. This allows Cable to save the future and the present.
Utilized (rather unfairly from some viewers' POV) in the short-lived Dragon's Lair cartoon. The show would often go into commercial breaks with Dirk facing an A or B choice. In the original video game, one would mean safe passage the other, instant, hideous death. In the cartoon, both meant death. But, as the narration would smugly inform us, "Dirk saw there was a better way".
Also used in the episode "Simian Says" where Mojo Jojo kidnapped the narrator and narrated the girls doing his bidding. When they found him out and asked where he wanted to be punched, stomach or head, he replied "How about an option of the third type?" He didn't get to take it.
Also used in "Not so Awesome Blossom," where Blossom has to decide between attacking Mojo and risk having the Professor fall to his death or accept him as her ruler. She ends up taking a third option in an unexpected way.
Nudar: You have two choices: unconditional surrender... Leela: Never! Nudar: Or total annihilation. Leela: Also never! Nudar: You have thirty seconds to decide. Leela:NEVER!
In the Space Ghost episode "Zorak", Zorak kidnaps Space Ghost's teen sidekicks and forces him to fight his giant hornets without his power bands, or his sidekicks will die. After fighting the wasps for a few minutes, Space Ghost puts his power bands back on and defeats the hornets, saving his sidekicks shortly after. The third option, if you missed it, was "remember that all your enemies are idiots," or "both" for short.
In a Teen Titans episode, Starfire's pet worm is torn in a decision between his father and Starfire, who raised him with love. Rather than joining either, he takes a third option, and explodes. Subverted in that this made him shrink down to baby size, where he happily stays with Starfire.
In the first season finale, Robin has to choose between serving as Slade's apprentice or letting a bunch of Slade's micro-probes kill his friends. His choice? Infect himself with the probes so that he and his friends share the same fate; Slade's so obsessed with winning that he'd rather let the Titans live than "lose" by allowing his apprentice to die.
When Homer shows Kang and Kodos are impersonating the 1996 presidential candidates on The Simpsons, they note the revelation makes no difference.
Kodos: It's true, we are aliens. But what are you going to do about it? It's a two-party system; you have to vote for one of us. Man 1: He's right, this is a two-party system. Man 2: Well, I believe I'll vote for a third-party candidate. Kang: Go ahead; throw your vote away!
The same episode has Kodos and Kang trying to tell the crowd what they want to hear. "Abortions for everyone!" gets booed, "No abortions for anyone" also gets booed. They settle on abortions for some and miniature American flags for others.
In another episode, Bart and Lisa stand on opposite sides of the living room and ask Maggie to walk to the one she loves best. Maggie's choice? The television.
In one episode, the King Solomon example from the Religion section is parodied when Homer dreams that he is King Solomon and has to solve the conundrum of two men claiming ownership of the same pie. His solution:
Homer/Solomon: The pie shall be split in two, and both men shall receive... death. I'll eat the pie.
American Dad!, "Bully for Steve": With his own father acting as a bully to try and toughen him up, Steve is left with two options. Get tougher, or face more bullying from Stan. Instead Steve finds Stan's old bully, Stelio Kontos, who promptly gives Stan a complete ass beating.
A later episode had Steve faced with a bully his own age. He tries to solve the problem by hiring Stelio again, but the younger bully fanboys out on the legendary Stelio and then they team up to give Steve an even worse beating (after recording a remix of Stelio's theme song).
Spacely: I could make you vice president, or give you stock in the company, but I've come up with a better idea. George: Oh, what's that, sir? Spacely: I'm not going to dock your pay, for the time you were gone from your job. George: Oh. Thank you, sir.
In the "Double Date" episode of Justice League Unlimited, the Huntress has the option to either kill Mandragora in front of his own son for murdering her father, or to allow them to escape unscathed to a better life overseas. She chooses to capture and arrest Mandragora instead, a far less traumatizing event for the aforementioned son.
In the original Justice League series, Aquaman must deal with being in a Death Trap alongside his baby son. The first two options are saving himself at the cost of letting his child die and freeing said child only, but dying in the process. What's his choice? Cutting off his own hand... which lets him get free and save both himself and his son. And become a Handicapped Badass.
The episode "Another Way" is all about this trope, with every conversation ending with Finn shouting "My way!" When a tree stump (don't ask) tells him that he can only take the "smell bad forever" path or the "hair falls out forever" path, he kicks the signpost down and charges through the thorny bushes between the paths.
In "The Maze", Finn, Jake, and four Hot Dog knights enter the maze to find a wish-granter at the end of it. Along the way, two of the four hot dog knights die, and Jake pushes his stretching ability to the absolute limits with fatal results. At the end, Finn and two of the knights make it to the end. The first knight wishes for a box, the second wishes to "blow up" (he meant grow bigger, but the guardian was very much a Literal Genie) and Finn's stuck between getting what they entered wanting to wish for (a two-headed psychic war elephant) or wishing the hot dog knights and Jake back to life. After some thinking, Finn wishes for the elephant and makes it wish for Jake and the knights to come back to life. Then they mount up on the elephant and fly out of the maze while the guardian screams at them for "cheating".
"Ponyville Confidential": The Cutie Mark Crusaders have become pariahs when everypony finds out that they were the ones damaging their reputations with lies in ''The Foal Free Press' under the pseudonym Gabby Gums. They feel guilty about it and want to stop writing slander, but editor Diamond Tiara doesn't want them to, since said slander is bringing in lots of money for her, and threatens to publish an embarrassing picture of them in the paper if they dare to. Seeing that both choices will leave them social outcasts, they come up with a solution: issuing a public apology to everypony for the damage they caused.
"Keep Calm and Flutter On": Discord, taking advantage of Fluttershy's deep desire to be friends with him, gets her to promise to never use her Element on him, thereby preventing the others from subduing him again (since all six elements are needed for them to work). He then breaks out the chaos, and presents Fluttershy with a dilemma: she can either sit and watch as he devastates Ponyville... or she can break her promise. Instead she simply declares that she doesn't want to be his friend anymore. Though he first mocks her for it, a second later he realizes he's feeling genuinely bad about it, and calls the whole thing off for her sake.
"Rainbow Falls" has a case where the third option fails to solve the problem because it's just trying to dodge the whole issue. Rainbow Dash has to make a Friend or Idol Decision over whether to join the Wonderbolts' relay team or remain with her own. She attempts to dodge it entirely by pretending to be injured so she can't fly for either team. It gets lampshaded by Twilight, who points out that Rainbow Dash isn't actually making the choice at all, and is rather choosing to avoid addressing the question of her priorities.
In King of the Hill, some Buddhist monks are looking for the reincarnation of their lama. The test to determine the new lama involves candidates picking out one item owned by the previous lama from a collection of other random things. Bobby passes by accident, and he needs to take another test to confirm whether it was a coincidence or not. Upon learning that lamas are celibate, he doesn't want to proceed, but his Buddhist girlfriend, Connie, says that he has to or she can never be sure if their relationship is against her religion or not. When presented with another collection items and told to pick whatever he wants, he notices a mirror with Connie's reflection in it and picks her, which the head monk says is a valid option. The episode ends with the monks privately noting that the mirror was, in fact, the correct item.
In another episode, Peggy is arrested for kidnapping a Mexican girl after a misunderstanding stemming from her poor Spanish skills. Of course Hank doesn't want his wife going to jail, but he's worried that if he just straight-up testifies that her Spanish is bad it'll hurt her ego. His solution? Convince Peggy to speak in her own defense; the judge witnesses her poor Spanish first-hand, realizes it was a misunderstanding, and declares her not guilty, while Peggy remains convinced that impassioned plea managed to sway him.
This is actually Doug's biggest skill: being able to find some sort of compromise to a situation.
One episode has our title character accidentally hit a member of the AV Club and, when his dad finds out, gives him the sage advice "Show me a man who resorts to violence, and I'll show you a man who has run out of good ideas". Afterwards, Doug finds himself harassed by the AV Club, egging him on for a rematch. When Doug has had enough, he confronts the kid... and finds out that he didn't want the fight - his clubmates pushed him into it! So, either Doug backs out and they both come out as chickens or they go at it and Doug, at most, becomes a bully. The solution: stage a fight in the AV Room, using a television feed and, at the right moment, kill the connection and let Skeeter make the sound effects!
Another episode had the student body attempt to invite their favorite band, The Beets, to their school to have a concert, but Mr. Bone refuses, hating their music and proclaiming that his yodeling is much more sophisticated. It takes an Imagine Spot starring his superhero self Quail Man for him to come up with a Third Option - let Bone's yodeling group be the opening attraction before The Beets. Mr. Bone still doesn't like the band, but he's able to have some sort of spotlight.
DuckTales (1987), "Launchpad's First Crash": While the men and women are willing to cooperate with one another to rescue Grunta, they spend some time arguing over whether the head woman or the head man should lead the expedition. Launchpad proposes a third option which everyone agrees with — he'll lead the rescue mission.
Subverted in an episode of Fish Hooks: Bea is working at a pretzel kiosk and her manager makes her decide between firing Milo or Oscar. At the end she tries to fire the manager, but of course she can't and the manager fires all three of them.
Deconstructed in TRON: Uprising. A terrorist ties Tron to one bomb, sets the other up at Able's garage, and tells Beck there's only time to pick one. Beck's attempt at a third option is to send Able to defuse the one at the garage. Not only does this get Able killed, but now Beck's friends have turned against the uprising and sided with Beck's enemies because they blame him for Able's death.
In the original Thunder Cats, there was something like this in the second part of "The Anointment of Lion-O". Outracing Cheetarah the standard way was impossible, and he knew it, but he was allowed to take a shortcut, and seeing as she couldn't run as long as he could, doing so gave him a better chance. The downside? It was far more dangerous. Still, he took that option and won.
It's subverted in another part of that storyline where Lion-O faces off against Wily-Katt and Wily-Kitt in a challenge of wits. At one point, Lion-O finds three tunnel entrances before him. Katt playfully urges him to take the Left tunnel while Kitt equally urges him to take the Right one. Lion-O defiantly takes the center tunnel instead, which is precisely the one they wanted him to take for their own purposes.
Parodied in an episode of Johnny Bravo: while running a poll for Assistant Litter Commissioner that the options are Johnny and Carl, a guy answers "I like pie!" The papers publish: Carl 99%, Pie 1%, Johnny 0%. And the pie is later brought to a debate!
SpongeBob SquarePants: In "The Krabby Khronicle", SpongeBob is forced to write humiliating lies about his friends in Mr. Krabs' new newspaper enterprise. His problems are 1) He feels guilty that the stories are destroying the lives of everyone he cares about, and 2) Mr. Krabs is threatening to take his frycooking duties away if he stops, since the paper is bringing him tons of money. SpongeBob's solution? He publishes an article about Mr. Krabs overworking him and being the actual mastermind behind the bogus news stories, which makes his customers angry with him and take all their money back.
In the Penn Zero: Part-Time Hero episode "I'm Still Super!", Rippen subjects Penn to a Sadistic Choice where he can only save either Boone or Sashi from getting killed by an overly-complicated death trap. Professor Evil Professor, deciding that Rippen has outlived his usefulness, tells Penn that both of his friends will go free if he destroys Rippen instead.
In the episode "Keystone Motel", Ruby and Sapphire argue over how to deal with Pearl: they can't just forgive Pearl's actions, because they hurt Garnet so deeply. But equally, they can't ignore the problem (because doing so hurts everyone else). Eventually they reach a compromise (signified by the two fusing back into Garnet): they'll stop giving Pearl the silent treatment, but they won't accept her apology yet.
In "Message Received", Peridot is trying to help the Crystal Gems destroy the Cluster that threatens to destroy the Earth, but she is torn between her friendship with Steven and her loyalty to Yellow Diamond, who created the Cluster. So she contacts Yellow Diamond directly and tries to give her logical reasons why the Earth should not be destroyed.
Over the Garden Wall: The Beast tells Wirt that he can choose between watching Greg turn into an edelwood tree and effectively die, or spend the rest of his life tromping around dark wood turning other edelwood trees into oil for the lantern, which will supposedly contain Greg's soul. Wirt instead opts to just grab an axe, cut Greg loose, pick him up and run like hell.
In one episode of Ella the Elephant, the island is hosting a soap box derby, and Frankie and Ella decide to work together on their kart. However, Frankie doesn't listen to any of Ella's design ideas, only using his own, causing her to quit the team. Frankie later reconciles with Ella by opting for a third option - a kart that is half Ella's and half Frankie's design.
Over the course of Regular Show's run, the viewer is left wondering whether Mordecai will get together with Margaret or CJ. In the show's final episode, he's shown married to an unnamed bat.
In the Courage the Cowardly Dog episode "Journey to the Center of Nowhere", the two leaders of the sentient eggplants are divided on whether they should fry Muriel or grill her, until another eggplant walks up and suggests they bake her.
In The Loud House episode "In-Tents Debate", Lincoln is put in charge of deciding where the family will go on vacation. Half of his sisters want to go to an amusement park while the other half want to go to a beachfront resort, and each side tries to win over his favor and sabotage the other. In the end, Lincoln decides on them going to the campground that they were originally going to go to.
In The Fairly OddParents! episode "Wish Fixers", Timmy signs up for a program by HP to get him to stop making bad wishes. Not only does he he completely fail to curb his streak, every time he makes a wish that ends in catastrophe Cosmo and Wanda are electrocuted with Shock Collars that are turned up higher with each use, and the highest setting will destroy them. HP gives Timmy a list of pre-approved wishes and the only wish on it is to sign over control of Fairy World to the pixies, which will result in Cosmo and Wanda being replaced with pixies, so Timmy is sunk either way. His solution: wish that Cosmo and Wanda were made of rubber, which is non-conductive, and since it was done for their safety he technically broke his bad wishing streak.