[response to Diceman’s hijack]
The last time the Mirror Universe was treated was in the DS9 episode “Shattered Mirror” which sounds like the one you mention; I haven’t seen it so I don’t know exactly what happened.
[/response]
–Cliffy
You caught a glimpse when Ardra brielfy impersonated Troi in an attempt to seduce Picard.
What about the one where the crew met the singularity (or was it a rift in the space-time continuum) and had the strange encounter with the other beings before they reversed the polarity on the warp capacitor to safely disengage from the conflict? Wouldn’t one of those be considered a double?
Captain Pike ended up with a double who would stay with Vina on the planet Talos in “The Menagerie” (TOS).
Data’s other brother is B4. Clever, eh?
Not really.
Pike didn’t leave a double. That WAS Pike on the planet; the Tallosians used their illusion-powers to make him think and feel that he was young again.
Although they never had the opportunity to meet, Abraham Lincoln had a double in the original Star Trek.
The following is technically correct, but probably not in the spirit of the OP, but the TNG episode where Worf bounces from universe to universe in the Multiverse (“Parallels”, I think) with changes both subtle (Data’s painting) and not (Worf was married to Troi in the last universe, and Picard was dead). Anyway, in one scene when Worf is heading back to his original Enterprise in a shuttle (and is attacked by the insane crew of the Borg-dominated universe Enterprise (Insane bearded Riker - “We won’t go back - you don’t know what it’s like there!”), there’s a shot with a bunch of Worfs in different poses as he crosses an anomonly.
Parallels
For some reason I find Wil Wheaton line about “We have 300,000 [whatever] hails” when all the Enterprises start popping in hilarious… hmm, I guess we have >300K Trois and Rikers and Datas and LaForges (but at least one less Picard) in that scene also…
How about when they took the duplicated Harry Kim because the real one had died?
Can’t quite remember the ep…
You’re right, I misremembered. In the original pilot episode, “The Cage”, the Talosians created a duplicate of Pike as a companion for Vena. When “The Cage” was re-edited into “The Menagerie”, Pike himself stayed on Talos with Vena.
They also took the duplicate of that baby (Naomi Wildman?)
Oooh, I just thought of a fun one: there’s a scene where Lwaxana Troi is roaming about the ship, and has to rely on the Enterprise’s computer to direct her on where to go.
There’s nothing remarkable about this, except that Majel Barret played both Lwaxana AND the voice of the computer. 
Wasn’t there a Voyager epsiode where a psuedo-Janeway appeared on the ship iwth some evil intent or other? The real Janeway was held hostage somewhere…?
Of course, there’s the unicorn dog in the Kirk transporter double episode.
And the opposit of the doube experience when Tuvok and Nelix merge.
This week’s new episode of Star Trek: Enterprise (“Exile”) has somebody turning his own image into the likeness of a few members of the crew.
The psychotic guy who took over the mental institution planet also turned himself into the likeness of Kirk – although his plot was foiled when Scotty said, “Queen to queen’s level 3, sir.”
Similarly, the last episode of ST:TOS – “Turnabout Intruder” featured a woman who swapped minds with Kirk and ran around in his body pretending to be him, because women weren’t allowed to be starship captains in the 23rd century.
Geordi created that holo-duplicate of starship designer Leah Brahms, became attracted to it, and later went after the real Dr. Brahms, who thought the whole thing was highly creepy.
Harry Mudd had an entire planet of androids, most of which were duplicates of each other. Not to mention the 500 copies of Mudd’s wife that the crew left behind to torment him.
Roger Corby and his assistant, Brown, were duplicates of the real Corby and Brown.
Kahless, of Klingon legend, had at least two duplicates: one made by the Klingons from a blood sample (TNG), and another created by the rock-beings in “Savage Curtain” (TOS).
As I understand it, the Talosians wanted to keep Vina happy, so they gave her an illusion that Pike had stayed. Vina lived with the illusion Pike until the mutilated real one came back and usurped the illusion.
That must have caused some confusion… after all, several years must have elapsed, during which Vina was having picnics with her illusion of Pike, and the real Pike was traveling around to schools teaching small children not to play around warped baffle plates! 
Well, Vina knew Pike was leaving, hence the “I can’t go with you” line. Introducing the real Pike, along with his thoughts and dreams and memories, might make for a broader range of fantasy, since I assume after 11 years, a fantasy Pike would get pretty repetitive.
Then again, the Talosians “assimilated” all of human history (or at least what was stored in the Enterprise’s computer) so they should have lots of ideas to keep Vina entertained.
Besides, he wouldn’t completely usurp the illusion. His scarred physical body remains trapped in that wheelchair thingie. His interactions with Vina will be exclusively through an illusory “avatar” of sorts, just as her scarred form will be concealed from him.
[Geek overload…]
Anyhoo, I don’t think anyone has mentioned the baby Spock from Star Trek V yet, an illusion created by Sybok’s telepathy (along with a younger Sarek and Amanda). Similarly, a illusory Leonard McCoy engaged in the mercy-killing of his father in that scene.
Not unlike Kirk from “The Enemy Within,” B’Elanna Torres was split in two (a human B’Elanna and a klingon B’Elanna) in Voyager’s “Faces.”
–Cliffy
I vaguely remember the episode, but was that the real reason for the switch? I thought it was something else.