1. Kirk saves Spock
In Leonard Nimoy's directorial debut, Star Trek III: The Search for Spock, Admiral James T. Kirk breaks Starfleet rules in a way that is more serious than just bending them — and the fans love him for it.
Kirk is not allowed to help when he finds out that Captain Spock's katra is in Dr. McCoy and that his friend's body might still be saved from the unstable Genesis Planet. Instead of taking that option, he breaks McCoy out of the detention center and steals the Enterprise from Spacedock.
The operation is successful, but Kirk loses his ship, his son David, and a peaceful future in Starfleet. Kirk transports Spock's body to Vulcan, where the dangerous fal-tor-pan ritual reunites body and katra, reviving his friend at considerable cost.
This goes beyond breaching the rules; it's Kirk's ultimate belief that some allegiances are more important than prestige, celebrity, or a famous spacecraft. Kirk's somber, practically broken response to the Enterprise's demise shows that he knew the sacrifices he was making but made them anyway.