4. Kirk takes Spock to Vulcan
In the Star Trek: The Original Series season 2 episode "Amok Time," Captain Kirk puts Mr. Spock ahead of the task for the first time and most strongly. Kirk should attend the inauguration on Altair VI because Starfleet ordered him. However, he learns that Spock is enduring pon farr, the Vulcan mating ritual that will literally kill him if he doesn't return to his home planet to perform the necessary rites.
Kirk takes the Enterprise to Vulcan to save his first officer. This violates orders and jeopardizes his career. Kirk even appears to die at Spock's hands in a ritual combat that tests their friendship before McCoy's deceit is discovered.
This episode is popular because it showcases Kirk's emotional leadership, details Vulcan culture, and shows just how far the captain of the Enterprise will go for his friends.