The Alternative Factor - S1-E28
Plot hole: Kirk knows that Lazarus is insane and that he wants the Enterprise dilithium crystals. Yet he's not restrained in sickbay and is, in fact, given free run of the ship so that he can knock out the crew in engineering and steal the crystals. Other than to further a woefully weak plotline, this makes no sense whatsoever.
Is There in Truth No Beauty? - S3-E5
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Even if he did, humans can still go mad if wearing one and that's why Spock operates the transporter. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.
Capt. Kirk: Space, the final frontier. These are the voyages of the starship Enterprise. Its 5-year mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man has gone before.
Who Mourns for Adonais? - S2-E2
Trivia: An ending that was planned but abandoned for this episode would have revealed that Lieutenant Palamas was pregnant with Apollo's child.
The Corbomite Maneuver - S1-E11
Question: At the beginning, Mr Sulu calls for Captain Kirk to report to the bridge, but Kirk in sick bay does not get that message. Why?
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.