I like all of the above suggestions and find them of merit. I'll add one that most will probably not mention from an episode that's not well-loved, the final moment of "Requiem For Methuselah."
After Kirk's wish to "forget" his recent ordeal, and McCoy's echoing of that sentiment, Spock takes it upon himself to grant his friend's request. It was a moving moment in the series, and one that I will never "forget."
I'm so glad you posted this. It's one of my favorites too.
There isn't one single greatest moment, for me anyway. But there are definitely many great moments.
I love that "Risk is our business" speech. It exemplifies so much of what sci-fi has touched on over the years. This is the speech Dr. Rush should have made to the crew aboard Destiny.
Kirk realizing the Horta is intelligent and deserves to be protected.
When Kirk tells the Vians how they've lost compassion, that they're nothing but intellect... and it dawns upon them that he's right.
At the end of "Balance of Terror", when Kirk extends a hand of rescue to the Romulan commander.
In Plato's Stepchildren, when Kirk informs Scotty, "prepare to beam us up. I have a little surprise for you. I'm bringing a visitor aboard." And Alexander beams a big smile.
"The Savage Curtain", when Kirk first welcomes Abraham Lincoln aboard. You can see how taken he is with this incarnation, remarkably realistic.
The end of "Who Mourns for Adonais?". McCoy laments how he wished they hadn't had to do this. And Kirk says "So do I. They gave us so much. The Greek civilisation, much of our culture and philosophy came from a worship of those beings. In a way, they began the Golden Age. Would it have hurt us, I wonder, just to have gathered a few laurel leaves?"
The ending of the Menagerie, Part II... when the Talosians show Kirk how Pike has found his place, and as they wish Kirk well in his journey, he pauses to smile. I could almost see Shatner's hope and promise for what lays ahead in the next season of Star Trek.