"Who Mourns for Adonais?" by Gilbert Ralston and Gene L. Coon
Ah, the 60s... "One day she'll find the right man and off she'll go, out of the service."
Anyway, the Enterprise is mapping a planet when, suddenly, a giant green energy hand appears and grabs the ship, stopping it cold. A giant face appears and calls them "my children." He threatens the ship with squeezing, putting actual pressure on the hull. He invites them to the planet (except Spock, who reminds him of Pan) and Kirk, McCoy, Scotty, Chekov, and Lt. Carolyn Palamas (specialist in archaeology, anthropology, and ancient civilizations). He introduces himself as Apollo. Of course, he likes Carolyn, and puts her in a lovely Greek-inspired gown. Scotty also likes her, so he (stupidly) tries to attack Apollo and gets smacked. Apollo also fries their phasers and cuts off communication. Kirk speculates that the beings we know from mythology simply appeared to be gods to the people of the time.
Carolyn and Apollo have a nice scene where he tells her the others have sort of faded, their energy returning to the Universe, and he's the last one left, having believed that humans would get here eventually. They are powered by love and worship. That's why he wants the Enterprise crew to live on the planet with him.
Observing Apollo being tired and needing to leave after expending energy, Kirk plans to have all 4 of the party come at Apollo at once, to try and exhaust him. Carolyn stymies this plan by intervening to "save" the party.
Meanwhile, Spock has Sulu looking for Apollo's power source, Kyle (first appearance) looking for a way to punch holes in the force field, and Uhura working on a communication workaround. They all get some nice moments, as does Chekov, who first appeared in the previous episode but didn't really have anything to do. Here, he starts on the Russian-centric running gag.
CHEKOV: He disappeared again like the cat in that Russian story.
KIRK: Don't you mean the English story, the Cheshire Cat?
CHEKOV: Cheshire? No, sir. Minsk perhaps, but...
His previous plan having failed, Kirk talks seriously to Carolyn, reminding her of her duty and responsibility and telling her to spurn Apollo. She lies to him, comparing him to a specimen for study, and he calls forth a big storm. Kirk tells Spock to fire on the temple and it's destroyed. Apollo, broken and defeated, calls out to the other gods that they were right, and fades away.
Lovely ending:
MCCOY: I wish we hadn't had to do this.
KIRK: So do I. They gave us so much. The Greek civilization, much of our culture and philosophy came from a worship of those beings. In a way, they began the Golden Age. Would it have hurt us, I wonder, just to have gathered a few laurel leaves?
I've always had a soft spot for this episode because I'm a huge nerd for Greek mythology. Michael Forest is marvelous as Apollo - capricious, by turns bratty and caring, and his tears at the end truly touched me.