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, "Good call—Help!"
Played with in Avatar: The Last Airbender, 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.
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 powers 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 a viewer's 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.
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.
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!
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.
In Tangled, Mother Gothel is about to forcefully take Rapunzel away, but Rapunzel promises she'll go with Mother Gothel willingly if she can use her hair to heal a fatally injured Flynn first. Flynn quickly takes a third option by cutting Rapunzel's hair before she can heal him, allowing Rapunzel her freedom while denying Mother Gothel from her source of eternal youth and denying himself a chance to be healed from his mortal injury. He dies, but then the Swiss Army Tears kick in and he's revived.
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 re-mix 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 Adventure Time 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.
"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, Bobby passes a test presented by a group of Buddhist monks (where an array of items are laid down on a blanket), and he is thought to be the reincarnation of the Lama Sanglung because he picked up the lama's cane. Before Bobby is confirmed to be the lama's reincarnation, though, he needs to take another test. Bobby learns that if he really is the lama, he and Connie can no longer date, but if he doesn't take the test, Connie will dump him anyway. When Bobby is told to pick an item on the blanket to confirm if he really is the reincarnation of the lama, he picks Connie's reflection in the mirror. The head monk accepts his choice (but doesn't tell them it was the lama's mirror).
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.
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.
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!