So many great moments. I agree with most of the choices already put out.
I don’t know that I can say it’s the “single greatest”, but there’s one brief moment from Space Seed that I’ve always loved: in the seconds after he just barely manages to beat Kahn off, and avert the destruction of the Enterprise, Kirk is nearly doubled-over the console, trying to catch his breath. Then he suddenly looks over his shoulder in panic, half-raising the pin he used as a club, as if Kahn might be back on his feet – ‘cause he certainly might’ve been!
What I love about this moment is that, even though Kirk won, we can see that he is aware of just how narrow an escape it was. He only just barely managed to knock Kahn out, and he did it by being more brutal than we usually see him. There’s no self-satisfaction, just a lot of adrenaline pumping. Kirk only matched his genetically superior foe by digging into a deeper reserve of desperation and will, and in that one moment we can see that the entire encounter with Kahn has taken a heavy toll on his nerves.