I’m curious as to how many, if any, episodes of the original five broadcast Trek series did not include a single appearance by that show’s lead actor, the Captain for that particular show. I.E., TOS episodes in which Kirk does not appear, TNG episodes without Picard (none), DS9 sans Sisko, Voyager episodes without Janeway, or Enterprise episodes without Archer.
Are there any? If so, which ones?
For future reference, Memory Alpha has categories that can help with this kind of query; they're grouped together as "Regular cast non-appearances".
Using the resources there, it seems there are no episodes of the first five live-action Star Trek series where the captain (or equivalent) does not appear:
There are instances of many other major cast members not appearing, but not the captains. Note that those lists do mention episodes in which characters only appear off-screen (i.e. heard dialogue only) or in an unusually brief capacity (in a dream etc), so it would also seem there are no episodes where the captains only make appearances of those kinds, either.
William Shatner as Kirk did not appear in the animated episode, The Slaver Weapon. Its the only time in the first block of Star Trek history (not including the new shows) where the Captain did not appear.
If you can accept The Animated Series as a continuation of TOS, then Kirk did not appear at all in TAS: The Slaver Weapon.
There are two episodes where the captain only makes a very brief appearance:
In TNG: Thine Own Self, Picard has a single line of dialog in the last scene.
In DS9: The Wire Sisko has only three lines.
Shatner is credited in all 79 episodes of The Original Series, but not in The Cage, so your call on that one. Obviously, Kirk was in Menagerie 1 & 2, the retelling of The Cage, broadcast in November of 1966. The Cage was not broadcast during the show's initial run (IMDB seems to be wrong on this), running on TV first in 1988.
Stewart is credited in all 176 episodes The Next Generation.
Brooks is credited in all 173 episodes of Deep Space Nine.
Mulgrew is credited in all 168 episodes of Voyager.
Bakula is credited in all 97 episodes of Enterprise.
According to IMDb.com, accessed 08/11/22.
Also, Memory-Alpha.fandom.com "The Cage"
As close as any episode of the live-action broadcast shows come is TNG’s Thine Own Self, where Picard’s only dialogue is one short line. Others have covered The Animated Series.
If you do not consider Mirror Universe doppelgängers impersonating the real captain to count, a large part of season 1 of Discovery qualifies. An interesting judgment call is “The War Without, the War Within.”
Captain Gabriel Lorca (who had secretly been an impersonator for the whole season) was dead, and for most of the episode, Admiral Amanda Cromwell is the commanding officer. In the final scene, she names Phillippa Georgiou the new captain of the Discovery. But, that is really her Mirror Universe counterpart, the Empress of the Terran Empire, in disguise. But but, Admiral Cromwell knew that when she put her in command.
In the Voyager episode “Living Witness,” Captain Janeway only appears as a hologram
Major spoiler: whose portrayal is not entirely accurate.
The Enterprise two-parter “In a Mirror, Darkly” might also qualify on technicalities, if you squint hard enough.
The Mirror-Universe version of Archer (a different person in a different universe) is the main character of these episodes, we also see Captain Forrest of the ISS Enterprise, and Archer does take over the ship and start calling himself “Captain Archer,” but this is a hoax and he is not the lawful Captain.
Another amusing possible technicality inspired by a comment is “Farewell,” the season 2 finale of Picard.
The new Borg Queen took La Sirena and flew off with it in the previous episode, so neither Jean-Luc Picard nor Cristóbal Rios is the commanding officer of any ship any longer. Nor does Jean-Luc Picard hold the rank of Captain. He is either an Admiral, a General, or an undocumented immigrant. When Q snaps his fingers again, most of the time-travelers reappear on the bridge of the USS Stargazer, but not its captain, Rios, who stays behind. Admiral Picard does not take over as commanding officer of the starship he is on, but picks someone else to.
Since I don’t see any mention of them yet, Lower Decks and Prodigy both have the (Acting) Captain appearing in every episode. Not every episode of Short Treks is set on a starship with a captain.
Some interesting pedantic edge cases:
The finale of Enterprise "These are the Voyages..." only has a holographic representation of Archer, as part of the holodeck simulation run by Riker. (Note this isn't a surprise twist or anything: the episode is explicitly framed as a TNG-era story that includes an ENT holodeck sim.)
"In a Mirror, Darkly" follows a mirror-universe Enterprise; the main-universe's Archer is referenced explicitly when mirror-Archer learns about him through Starfleet records, but the character does not appear. (The captain of mirror-Enterprise does appear throughout, though. Whether you count this as an "appearance by the captain of the show's primary vessel" is a matter of whether you think the alternative-intro'd mirror-Enterprise show itself is still a show about main-Enterprise or mirror-Enterprise.)
Honorable mention to Voyager's "Course: Oblivion" which...
follows a duplicate Voyager, made of decaying exotic matter, trying to find the real Voyager. (Not a proper answer, however, because the real Janeway does make a very brief final appearance.)
Another pedantic edge case. The first 71 episodes of Deep Space 9, from S01E01: Emissary to S03E25: Facets do not technically feature the show’s main captain. Benjamin Sisko was introduced with the rank of Commander and held this rank until S03E26: The Adversary where he was at last promoted to Captain, a rank he maintained until the end of the series.
