Great responses on this topic. I agree with most of them and would be hard-pressed to choose a first place contender. I often think on subsequent rewatches that these famous scenes and speeches of Kirk aren't accurately rendered in clips and in small excerpts out of context. In "Return to Tomorrow," the intensity of "risk is our business" makes more sense if you understand (or buy into) the conceit of the episode: the potential dangers they faced and the tremendous trust Kirk was placing in these alien beings by allowing his and others' bodies to be completely occupied by a force against which humans had no control. He has to convince his own command team to agree with him -- Kirk even states that ALL must agree or none would participate (this is a detail I didn't notice until I was older, for some reason). Bones appears completely opposed and presents solid arguments against the whole idea; Scotty seems doubtful: even Spock seems more reticent than usual about the whole thing. Until the speech, of course.
In addition to all the great TOS episodes and scenes cited in earlier posts, let me add one other that I have recently grown to appreciate: during "Amok Time," Kirk/Shatner's confrontation and conversation with Spock in his quarters is masterful. The tension between the two is palpable. Kirk shows restraint and respect for Spock's sensibilities by choosing his words carefully and inquiring delicately... Shatner delivers such a convincing performance as someone holding back on his own natural impulses to help, to fix, to understand, to solve the problem for his friend. There is purposeful use of eyes contact, hand and body position, and precise physical movements that mirror the content of their uncomfortable conversation. And Shatner and Nimoy play off each other so well... we must acknowledge that Shatner's great moments often occur opposite Nimoy, whose own performances in TOS continue to amaze me.
In addition to all the great TOS episodes and scenes cited in earlier posts, let me add one other that I have recently grown to appreciate: during "Amok Time," Kirk/Shatner's confrontation and conversation with Spock in his quarters is masterful. The tension between the two is palpable. Kirk shows restraint and respect for Spock's sensibilities by choosing his words carefully and inquiring delicately... Shatner delivers such a convincing performance as someone holding back on his own natural impulses to help, to fix, to understand, to solve the problem for his friend. There is purposeful use of eyes contact, hand and body position, and precise physical movements that mirror the content of their uncomfortable conversation. And Shatner and Nimoy play off each other so well... we must acknowledge that Shatner's great moments often occur opposite Nimoy, whose own performances in TOS continue to amaze me.