The Enterprise is assigned to repair the Argus Array, a space telescope that has stopped working. (It's the 24th-century equivalent of the Hubble Space Telescope; was this story about a critical scientific tool in need of crucial repairs ripped from the headlines of the time?) A mysterious probe orbiting the array zaps Barclay while he's on a shuttle mission. After returning to the ship, Barclay has a newfound confidence and his brain activity increases exponentially. He becomes smarter and smarter, and that begins to worry some people.
The character outline is Flowers for Algernon, except instead of taking a mentally challenged man and turning him into a genius, it takes a man of average intelligence (for this crew) and turns him into an ultra-confident, cosmic super-genius. In the opening scene, regular-Barclay is playing Cyrano de Bergerac in a performance that, let's face it, is pathetic despite his best efforts. Later, watch how genius-Barclay's acting is so mesmerizing that it practically makes Crusher weep. Dwight Schultz's performance as Barclay is pitch perfect because it finds the right balance between earnest sincerity and dryly ironic narcissism. Schultz, and the episode, know that deep down this is all kinda funny because it's about Barclay, and they don't shy away from the comic notes of Barclay's growing ego and arrogance, even if he's always well intentioned.
Meanwhile, the imaginative sci-fi machinations proceed at warp speed. To fix the array, Barclay comes up with a brilliant plan that's impossible to execute by anyone except him, and requires a computer interface far faster than anything available, so he uses the holodeck to build a device that taps him directly into the ship's computer core (this device is both creepy and really cool; kudos to the production designers), where his brainpower expands and eventually takes over the entire computer and, thus, the ship. Barclay begins to develop a god complex, perhaps not unjustifiably, and claims he can understand the entire universe as a simple equation. He starts to scare the hell out of everybody.
The way the crew reacts to all this is absolutely honest human nature; they fear what they cannot predict or understand, and I don't blame them — especially when Barclay puts an energy field off the starboard side of the ship and prepares to send the crew 30,000 light-years through it, while assuring everybody, "Please, you must trust me." The suspense of what waits at the other side is one of the true moments of unpredictable awe in the Trek canon.
What actually waits there, alas, cannot live up to that awe, but I did still enjoy the episode's sense of whimsical curiosity, in which it turns out that advanced aliens used Barclay as an implement to bring the Enterprise here in carrying out their own exploration of the galaxy. Barclay is of course returned to normal, which begs the question of whether it's a blessing or a tragedy to allow the blind man to see before taking it away again. "The Nth Degree" is a splendidly unique amalgam of tones and themes, plot and characterization, imagination and bemusement, and it ends up being one of the most fascinating hours in TNG's run.