This is episode 6 of the first season, and aired on October 26, 1987. The title is a fannish nod to an episode
from original Star Trek; 1966's "Where No Man Has Gone Before." That episode was the second pilot (after "The Cage"), and also features super-human powers and the edge of the known galaxy. "Where No One Has Gone Before" looked interesting when I first saw it, all those years ago, because the screen play was by Diane Duane and Michael Reeves. Both had written a number of Star Trek novelizations for the original series. Duane is a top notch writer, with talent not only in terms of the Trek novels she penned for Pocket Books (who own the Paramount Trek tie-in franchise) but a fine writer on her own merits. Indeed, the basic plot for the episode borrows heavily from Duane's 1983 Star Trek novel, The Wounded Sky.
The basic plot line is this: Star Fleet sends a propulsion expert, one Kosinski, with his alien "assistant" to make changes to the Enterprise's engines. K has been credited with amazing results elsewhere, though the Enterprise Engineering team views his suggestions for changes with scant favor, pronouncing them nonsense. I note for the curious, that naming the chief Engineer Argyle is absolutely a fannish nod at Scotty, or maybe at Niven's The Mote in God's Eye. And, indeed, they are nonsense; the changes witnessed on the Enterprise and elsewhere, are largely due to the efforts of Kosinski's alien time-traveling assistant, known simply as The Traveler. Wesley Crusher, now firmly ensconced as a boy wonder, is actually likable in this episode, a credit to both Wheaton, and the writers. Wesley, allowed to observe the changes being effected in the engineering section, notices that while the Enterprise has unprecedented, miraculous speed increases, the alien Traveler appears worn out, even exhausted, and is seen first by Wesley, and later by Riker, to "phase," appear and disappear randomly. Similarly, the ship has managed to go much farther than intended or even desired. As the Enterprise tries to return home, the traveler, seriously weakened by his efforts, tells Picard to take care of Wesley; he is a prodigy, and preternaturally gifted. It turns out that thought shapes reality, with potentially disasterous results. There's a point where Picard opens the door of the turbolift and is about to step over the threshhold into a sky full of stars that's a nod at Duane's fantasy The Door Into Summer.
If you want the whole gory plot summary, go read this plot summary, but if you want something smart and funny, check out Wil Wheaton's review of the episode.
This was very loosely inspired by Duane's novel; and in fact, she and Reaves, while they wrote the initial treatment, wrote it before the characters, including Picard, had been cast, and then, the script was turned over to someone else entirely (who was later fired) for a re-write. You can read Duane's rather amusing take on the experience here, complete with earlier drafts. I gotta say, the performance that really stands out here for me is the actor (Stanley Kamel) who played Kosinski. The actor is someone whose face you've seen in lots of other roles, but he nailed the performance of an officious idiot without going over the top, no easy feat. Notice too, that the characters of the crew are really beginning to jell, particularly Picard and Riker, played by Patrick Stewart, and Jonathan Frakes. Both actors do a marvelous job of working in a team, without being subsumed.
Trailer for "Where No One Has Gone Before"
The previos episode was "The Last Outpost"; the next episode is "Lonely Among Us."

