The Corbomite Maneuver - S1-E11
Continuity mistake: While beaming to the alien ship, McCoy has his hands on his knees. When materializing, they are beside his thighs.
The Corbomite Maneuver - S1-E11
Revealing mistake: In Kirk's quarters, as he touches the desk top comm panel, it noticeably slides a little rather than being fixed in place.
The Corbomite Maneuver - S1-E11
Continuity mistake: As the bridge shakes after the object is destroyed, Bailey's chair goes from tilted to upright between angles.
The Corbomite Maneuver - S1-E11
Continuity mistake: The cube holding the ship in place is spinning counter clockwise, from left to right, on its axis. When the camera is looking at the bridge, the colors are going from left to right, rather than right to left to match the cube's movement.
Continuity mistake: The landing party beams down in daylight, but the window inside Mendez's office shows that it's night outside. In the next exterior shot, it's day again. Next shot back in the office: night. (00:00:30 - 00:05:20)
Other mistake: While on Starbase 11, Kirk and McCoy watch Pike on a starbase desktop monitor as his light keeps blinking no. When Kirk is with Mendez in his office discussing the Talos IV top secret file, Piper watches Pike on the desktop monitor in the office. Then Spock uses the monitor in his quarters to watch Kirk. All these monitors have the same dings and markings around their screens, which are also the same as other desktop monitors on Enterprise in the previous episode, The Corbomite Maneuver. (00:11:10)
Continuity mistake: Kirk introduced Spock as "Lieutenant Commander" instead of "Commander".
Plot hole: Apparently there is some confusion over the distance between Earth, Starbase 11, and Talos IV. When Spock first meets Pike on Starbase 11 he tells Pike Talos IV is only six days away. Yet when Pike (in the recording) speaks to the Talosians for the first time, he says he is from a star system on the other side of the galaxy. If Talos IV was on one side of the galaxy and Earth was on the other side, it would take hundreds of years at maximum warp to travel from one planet to the other.
Continuity mistake: As Veena (as the Orion Slave Girl) dances, the water in the pool is still, then in the Observation room, as Kirk and party watch, the water is disturbed. A moment later, it's flat calm.
Deliberate mistake: At the very end, the Talosians send a final visual transmission of Vina and Christopher Pike, now whole and happy and reunited after 13 years, holding hands as they enter the Talosian elevator in the hillside. However, in this last shot, the elevator is still half-disintegrated, exactly as it was 13 years earlier when the Enterprise crew destroyed the hillside with a laser cannon. Within the context of "The Menagerie" storyline, this suggests that the Talosians never attempted to repair the elevator for 13 years (even though they continued using it). This incongruity is due to Gene Roddenberry cannibalizing his Star Trek pilot "The Cage," which contained zero footage of Jeffrey Hunter and Susan Oliver entering the intact elevator together (only the destroyed elevator). So, Roddenberry deliberately tried to "slip one by" the audience in this brief shot.
Suggested correction: There are reasons why the elevator would appear damaged. As the Talosians were in control of everything shown on the ship's viewer, the entire scene could be an illusion, or at least the elevator's condition may have been, with the Talosians choosing to allow the viewers to see the elevator in the same condition they last saw it. Just as likely, however, is that the Talosians truly never did reconstruct the elevator, as the whole point of their having a menagerie of other beings was an attempt to breed a race that could physically serve them, for their concentration on their mental powers had led to a complete inability and unwillingness to perform physical tasks (like repairing an elevator).
Still, as long as the Talosians are creating the illusion of Christopher Pike and Vina in their "restored" bodies, why not create an illusion of the elevator and hillside restored, as well? One big illusion of restoration, rather than a composite of dismal reality and happy-ending illusion? Again, to the point of my original post, the obvious incongruity is due to Roddenberry using the only happy-ending footage he possessed, that of Pike and Vina entering the half-obliterated elevator as they did at the end of "The Cage." Certainly, if Roddenberry only had the foresight to shoot Jeffrey Hunter and Susan Oliver entering the intact elevator, he would have used that footage instead. Any attempt to explain away the 13-year incongruity is mere wishful thinking.
This would qualify as a question, not a mistake. It is entirely plausible that the Talosians wouldn't bother to repair the elevator. It's also possible, as the previous correction points out, that the entire scene is an illusion. Remember, Captain Kirk sees Vina and Pike together on the planet literally moments after Spock wheels Pike out of the room. It's unlikely Pike had already been beamed down.
Revealing mistake: When the Talosians place Christopher Pike and Vina into the "picnic" illusion (in the countryside on Earth), Pike wanders around marveling at how real it all seems. Well, "real" except for the fact that Pike's body is casting 4 distinct shadows in 4 different directions on the ground, the result of studio set lighting.
Other mistake: When the transporter comes on to beam up Pike, the blonde ensign goes to face the transporter. As she does, she glances down at the floor to find her mark, and stands on it, looking into the chamber.
Revealing mistake: The shot of the castle on Rigel VII seems to be what Pike sees through his eyes, but in the next shot he himself is in the shot.
The Conscience of the King - S1-E14
Revealing mistake: As Kirk and Lenore leave the cocktail party to walk around outside, they stop to talk. From Kirk's perspective, he is to the left of Lenore, enough to see what should be in view ahead of him. Only seconds later when she leans in to kiss him does he notice Dr. Leighton lying next to some rocks. She was not obscuring his view before. Why didn't Kirk see this while they first started talking? (00:10:36)
The Conscience of the King - S1-E14
Plot hole: After Kirk sees Leighton laying next to the rocks, he rushes over to examine him. He places one hand on Leighton's arm and the other on Leighton's waist and declares "It's Tom Leighton...he's dead." All this without the use of a medical tricorder or even the experienced opinion of the ship's doctor to confirm what Kirk is seeing. (00:10:46)
The Conscience of the King - S1-E14
Continuity mistake: When Spock explains to McCoy his theory of Kodos' use of eugenics, the background behind Spock is a wall of red grating, but the shots of Spock speaking to McCoy alone (McCoy isn't in the shot) show a blue background wall with a control panel on it. (00:21:55)
The Conscience of the King - S1-E14
Continuity mistake: After Kirk introduces himself to Lenore at Leighton's cocktail party, when she tells him, "Karidian is a most unusual man... That was my father," Lenore holds the goblet in her left hand as she takes a sip, but next shot the goblet is suddenly in her right hand as she sips.
The Conscience of the King - S1-E14
Visible crew/equipment: When Lenore arrives at the Leighton's cocktail party, there is no furniture behind Kirk in the center of the room where they will stand in a moment. When Kirk introduces himself, in the next three shots facing Lenore, in front of her arm we can see a bit of camera equipment at the bottom of the screen.
The Conscience of the King - S1-E14
Continuity mistake: After the phaser explodes in the pressure vent disposal unit, just before Kirk walks into Karidian's quarters there's a crown and board on the table beside Karidian, but when Kirk gives the paper to Karidian for him to read, the board is suddenly on the opposite side of the table, though it has not been touched.
The Conscience of the King - S1-E14
Continuity mistake: As Kirk asks the Enterprise's computer about the history of Anton the Actor and Kodos the Executioner, his hands change positions from being away from the computer's controls to near them, then clasped as the shots quick cut.