Many of the moments mentioned upthread seem to be the final lines of episodes: "...Forget," "...a few laurel leaves," "...bringing a visitor aboard," "Let's get the hell out of here," etc.
This is as it should be. All's well that ends well, or, in modern parlance: Satisfying Ending Is Satisfying.
I understand all the arguments against the "...Forget" moment, but it works for me every time, because of how it's handled: with the phrase from George Duning's "Empath" music hovering in the background, the soft gong as the strings pause briefly and then return with the volume slowly reduced so as to emphasize Spock's voice when it comes in - "...Forget" - and then a quick cut to the ship exiting, credits, etc. One of the very best pieces of work by the film editor and the music editor in the whole third season, to be sure.
My choice for an off-the-beaten-track great moment is Miranda Jones' line in the ship's arboretum, in another Duning-scored episode, "Is There in Truth No Beauty?":
"I've never been to Earth."
To me that has always conveyed a lot: Meeting a human who we know isn't originally from Earth conveys something about the Trek universe that hasn't been conveyed by all the various alien species we've met, nor by humans we know to be Earth colonists. Humanity has spread out so much that someone like Miranda can be a second- (or third-)generation resident of some other planet.
(Please correct me if I'm wrong, but I don't think any other human in the original series was known
not to have been from Earth.)