Mirror, Mirror (takes place almost entirely on board the Enterprise)
Terrific, evil twin fantasy. And Kirk gets a "Captain's woman," whose role in this universe is very frankly displayed.
Journey to Babel (ditto)
Kirk was never more heroic than when piloting the Enterprise through an alien suicide attack while nursing a stab wound in the lung.
The Corbomite Maneuver (ditto)
Bluffing our way out of disaster; in the depths of the crisis Mr. Spock almost says "I'm sorry," but manages to restrain himself.
Wink of an Eye (ditto)
Another sheer fun fantasy -- imagine if you lived at another speed than everybody else. And this time Kirk's woman gets to tell a jealous rival, "Allow me the dignity of liking the man I select." A rare gem from the otherwise notorious third season.
Balance of Terror (ditto)
We meet the Romulans, and see the Enterprise and crew functioning in a purely military situation. Refreshing.
Charlie X (ditto)
Kirk as father to a telekinetic teen spaceman. Only he could pull off (so to speak) the red tights.
The Galileo Seven
Spock gets most of the camera time, Kirk comes close to crying -- don't miss it.
City on the Edge of Forever
Great story, but Spock actually has the more interesting role.
The friendship of the three main characters is shown at its best. And you get to meet T'pau, the grand old lady we all want to be. Heck, T'pring is the grand young lady we all want to be.
Another great story -- how on earth are they going to disprove the evidence of the damning tape? -- in which Spock's role, again, is the most interesting.
Errand of Mercy (okay, let's make it eleven)
Kirk ends up embarrassed at his eagerness for a good war with the Klingons, though of course he is in the right -- and you have the fun of spotting the actor who played the valet to Maurice Chevalier's Uncle Honore in Gigi.
The Menagerie (it'll have to be an even twelve)
For sheer ingenuity, this one is hard to beat. Once more, Spock's role -- his total loyalty to two captains -- is the crux of the story.
Honorable mention: Plato's Stepchildren. Though painful to watch, nevertheless it does make you marvel. For one thing, maybe leisured life in ancient Greece was pretty grotesque in some ways. For another, these guys, as actors, had guts and they earned their pay. For this one they pretended to be puppets on strings -- they danced jigs -- Shatner had to crawl on all fours and whinny like a horse. And the script writers, to give them the benefit of the doubt, perhaps were thinking it was high time the heroes of the Enterprise acted heroic in some other way than the physical. There is a heroism, after all, in submitting to humiliation and remaining human, disciplined, and civilized in spite of it. The only flaw in the rendering of the characters' reactions lies in Spock's devastation at what he is put through. To be logical would have been to accept that the treatment meted out him to is not his doing and is therefore not to be agonized over.
Image from Memory Alpha