12. Tomorrow Is Yesterday
In the first adventure of Trek canon involving time travel, the Enterprise is hurled back in time to 1969 where it is detected by Earth satellites as a UFO. Fighter pilot John Christopher is scrambled to investigate but his fighter jet is accidentally destroyed by the Enterprise tractor beam, and Captain Kirk beams Christopher aboard to save him. Realizing that he has drastically altered the timeline and quite possibly destroyed the future they came from (it's all wibbly wobbly, timey wimey... wait, wrong franchise; sorry), Christopher must stay on board to avoid further contamination. Spock discovers, however, that between footage of the Enterprise picked up on video and the fact that the fighter pilot's unborn son will play a critical role in history, further contamination is rampant. Scotty is the first to postulate the theory of a slingshot around the sun (one that we will see twenty years later in Star Trek IV), a theory that by taking the ship around the sun at high warp with precise calculations they can reverse time to undo the damage caused by the fighter pilot being aboard and subsequently warp back to their own time. The slingshot is successful, the fighter pilot is returned to the moment before he discovered the ship and calls it in as a simple UFO sighting.
11. Arena
Oh yes, who could forget about this one? It's been lampooned on practically every single parody of this great show ever made (including a recent video game commercial with Shatner himself) and it is practically the poster child of why low budget in the 60s was pretty hilarious. While the Enterprise gives chase to a Gorn vessel which destroyed a Federation outpost, the two vessels invade the space of the Metrons who pit the two ship captains into a rather hilarious fight to the death including slow moving punches, makeshift gun barrels and overly dramatic boulder throws and an overall an hour of 21st century laughs and cringes.