According to the classic movie Star Trek II: The Wrath of Khan, James T. Kirk has never faced a no-win scenario. Except that anyone who's ever watched the original Star Trek knows that's not true. He faced them all the time. Here are 10 no-win scenarios that Kirk faced.
Before we get started, let's acknowledge that Nicholas Meyer, writer/director of Star Trek II, was not steeped in Trek continuity and actually hadn't seen much of the original series. Kirk certainly faced death and loss on many occasions, and several of his victories were either miraculous or probably short-lived. Or somewhat pyrrhic. This list draws on the Animated Series and tie-in media to some extent, but is mostly Original Series.
With that out of the way, here are the 10 times Kirk actually faced a situation where there was no "winning":
1) Tarsus IV
This is from Kirk's childhood, but it's still a situation he faced. When Kirk was a child, he was a colonist on the planet Tarsus IV, where a fungus destroyed most of the colony's food supply. The governor, Kodos, decided to seize power by declaring martial law — and then divided the population into two groups, executing half the colony's population so the other half could survive. As Spock explains in the episode "The Conscience of the King," "Relief arrived, but too late to prevent the executions." Kirk had to stand by and watch half the people on the planet being executed.
2) The U.S.S. Farragut
Another formative experience for Kirk. His first tour of duty as an officer on the U.S.S. Farragut, the ship encountered a mysterious cloud creature that slaughtered the captain and almost the entire crew. Kirk blamed himself because he hesitated briefly before firing his phaser at the creature — but in the episode "Obsession," Kirk realizes that phasers are ineffective against this creature, and in fact there was literally nothing he could have done to save the Farragut's crew.
Taking a brief swerve to the new movie continuity... in a universe where Kirk wasn't a colonist on Tarsus IV or an officer on the U.S.S. Farragut, he still faced a huge loss that he couldn't do anything about. The entire planet Vulcan gets destroyed in the 2009 Star Trek film, despite Kirk's best efforts to save it. Kirk isn't in command yet when this happens, but it still takes place on his watch and he has no way to save it.
4) Gary Mitchell
This one is slightly more debatable — after all, Kirk saves the Enterprise in the episode "Where No Man Has Gone Before." But he can't save his best friend Gary, who's turned into a superbeing after passing through an energy barrier. In fact, the whole episode consists of Spock convincing Kirk, over and over again, that the situation is hopeless and there's no way to save his friend and his ship. He has to choose.
5) The Salt Vampire
Meanwhile, the episode "The Man Trap" ends with afairly clear victory for Kirk — except that he wipes out the last member of an intelligent species, who had been living for years without harming anyone.
6) The Organians
When the Federation comes to the brink of war with the Klingons, the godlike Organians decide to step in and put a stop to it. Sure, this means Kirk and his crew get to go on living, and the galaxy isn't subjected to a brutal war, so that's sort of a win. But Kirk seems pretty pissed about it, asking the Organians "What gives you the right?" Later, Kirk confesses, "I was furious with the Organians for stopping a war I didn't want." If the Organians hadn't stepped in, the Enterprise might have been destroyed, but then again Kirk could have helped score a real victory against the Klingons. There are lots of episodes where Kirk outwits godlike aliens or shows them that humanity possesses greatness... but this isn't one of them.
7) Edith Keeler
Similar to Gary Mitchell, this is a situation where Kirk saves the Enterprise (and fixes the timeline) — but he can't save the person he really wants to save. Once again, it's up to Spock to make Kirk realize he can't have it all. He has to choose between Edith and everything else. All of the dramatic weight in this terrific episode comes from Kirk realizing that sometimes you can't pull off a miracle.
In the episode "A Private Little War," Kirk is faced with an impossible choice. The Klingons are arming one side in a local conflict with superior (albeit still primitive) weaponry. So Kirk decides to arm "his" villagers with equal weaponry, to level the playing field. Except, as McCoy points out, "that means you're condemning this whole planet to a war that may never end. It could go on for year after year, massacre after massacre." Add to that the weirdness that Kirk is possibly drugged out of his mind by his friend's wife. The whole episode is clearly meant to be a metaphor for the Cold War, but it has a super unclear ending. Kirk instructs Scotty to manufacture 100 flintlocks, but then changes his order to 100 "serpents, for the Garden of Eden." The episode's final words are "beam us up, Spock. We're very tired." And that's... it. Did they supply the 100 flintlocks, or did Kirk just decide it wasn't worth it? Either way, he's basically faced a situation where he can't really win.
Now we're starting to reach into tie-in media somewhat. In the episode "The Apple," Kirk discovers a primitive society feeding a dictatorial computer with a lizard head. And Kirk destroys the computer, frees the people, gives them a few minutes of sex education, and then leaves. In a 1987 comic book, we revisit the planet and discover that Gamma Trianguli VI was a synthetic planet created by the survivors of a planetary disaster, who were determined to prevent their own tragedy from recurring — so they created Vaal to serve as caretaker of the world. And once Kirk destroyed Vaal, everything went to hell and the people on the planet set about destroying the entire world again. In the comic, Spock risks his own life to recreate Vaal, to save the planet. Because it turns out that Vaal was necessary, and blowing him up wasn't the neat solution Kirk thought it was.
10) Dramia II
And finally, an episode of the Animated Series — the Enterprise crew catches a space plague in the episode "Albatross," and only Spock is unaffected. Kirk puts Spock in command of the ship, and before he leaves he issues General Order VI. This calls for the ship to wait until the last human aboard dies, wait 24 hours, and then self-destruct. To prevent anyone else catching the disease. McCoy finds a cure at the last possible minute — but Kirk was ready to blow up his ship and admit defeat.
Many thanks to our Star Trek brain trust: Greg Cox, David Mack, Kevin Dilmore and Robert Meyer Burnett. You guys were a huge help!
Screenshots via TrekCore.