I was trying to figure this out the other day.. we used to watch VOY's Future's End every time we were having movie night at home because it was more fun then most movies. I've seen it more times than my favorite VOY ep, Timeless. So there's the approximately 4 times all the way through VOY viewings and then the movie night events for Future's End. My favorite ENT ep, E2, I've only seen 4 times, during each ENT rewatch, never chose to watch it separately. This is what happens with eps that take place in a bigger story arc, I don't get around to watching them as stand alones. None of these eps are the winner though. When I was a young teenager I was watching TOS twice a day, five days a week because I could get it on two different channels by skilful antennae twisting. I had a little notebook with the ep descriptions cut out of TV guide and pasted in, I would put a tick next to each ep as I watched it. The eps were shown randomly yet there was definitely a person choosing them. The ones considered great were shown far more frequently whereas there were eps like The Lights of Zetar which I only saw 3 times. City on the Edge of Forever was the clear winner in the notebook with 13 or 14 viewings and since I've seen it at least 5 times since then it is the winner.
I understand your enthusiasm. Prior to VCRs and DVDs I would write down the plots of each episodes and my favorite moments. My winner would have to be "The Best of Both Worlds, Pts. I and II". I have used those episodes to gather strength, as I watch Riker get stronger after the death of Picard.
I'd have to say from TNG probably "Yesterday's Enterprise", I just love that whole story. I love time travel/alternate realities stories. But I also love "The Best of Both Worlds" 1 & 2 and always watch when they are on. I have seen them many many many times. And for ENT, definitely "Shuttlepod One", my favorite in all of Trek. Love to see the growing friendship of Malcolm and Trip. "The Communicator" is another I have seen many times. If for no other reason because it's a Malcolm episode.
I'm honestly not sure, but it would have to be a TOS ep, just by virtue of all the years those aired in reruns before any other Trek even existed. Some episodes seemed to air a little more frequently than others - possibly one like "Shore Leave" or "Amok Time" or "Tribbles"?
TNG season 1 was the first Trek season I had on DVD, and it was the only one I had for a while, so whichever episode I've seen the most probably comes from that season. If I had to guess it'd be "11001001". That's probably my favorite from that season.
Over all series, including films, it's The Doomsday Machine for me. For non-films and non-TOS, it would be The Best of Both Worlds (part 1). For films, it's The Wrath of Khan; I played my tape until it wouldn't play any more.
When I need some cheering up, I usually watch either Amok Time or Trials and Tribble-ations, so I have seen these episodes quite frequently. But I've also watched endless reruns of TNG during college.
I think the episode I've seen the most times is "The Measure of a Man". It's one of those great Trek classics that I can never get enough of.
The Cage Amok Time Return To Tomorrow Cause & Effect (love a causality loop) BoBW YOH Living Witness Pale Moonlight In A Mirror Carbon Creek
"The Way of the Warrior". It was the first Star Trek VHS release I ever bought and also the first novelization.
TOS-"Where No Man Has Gone Before" I never get tired of seeing it, but it has been at least a good four or five years since the last time I did.
"Tribbles". Or "The Menagerie". Or even "I, Mudd". Those are the episodes I've had on home video the longest.
True Q comes to mind because whenever TNG is on TV, it seems 80% of the time that episode gets played. Otherwise, The Doomsday Machine is almost certainly my most watched episode.
I don't even know, probably either City, Crossfire, The Magnificent Ferengi or What You Leave Behind.