The Paradise Syndrome took place over almost a year.
This one's easy: Spock tells us that the time-consuming part of the episode, the slow coasting of the ship back to the planet apace with the giant asteroid, took exactly 59.223 days. He also tells that the trip to the asteroid at the beginning took "hours", and that the teaser and adjoining search of Kirk was limited to 30 minutes max. And for the final events of the episode, he specified that the asteroid would arrive four hours after the ship did.
So we get an exact duration of sixty days, no more, for the entire adventure. Unless we assume that Spock's 59.223 days were at very high sublight speed (after all, the reverse journey had taken "hours" at speeds up to warp nine), and that time dilation meant that a lot more time would pass for Kirok down on the planet... But there's a weakness in that: McCoy also estimated a two-month journey earlier on, and he'd be speaking from not just the layman's point of view (even laymen might get relativity when it affects their everyday lives), but from the planetside point of view. He clearly says the planet has two months till the asteroid arrives!
So, sixty days. Some nondescript episode like "Explorers" might easily log more than that.
And my candidate for the episode spanning the longest time interval would be "Death Wish", since it's the only episode where our heroes go all the way back to the Big Bang (even "All Good Things.." only goes a few billion years back). But within the rules of "the characters must have lived through all of it, no reset button", I must agree that "The Expanse" gets some sort of a prize. Although those rules might allow for "Blink of an Eye" as well, since several important characters experience decades - and their culture centuries - of very real passage of time which is not reversed for them. A main character adds two non-reset years to his life, even.
Although in the latter respect, Picard's added decades from "Inner Light" and O'Brien's jail sentence from "Hard Time" (neither of which was negated psychologically, even if neither aged the heroes physically) get the two top prizes. The EMH's stay on the "Blink" planet wins in a different category, though, as he did age physically.
Timo Saloniemi
