Continuity mistake: In season 6 episode 10 'Pathfinder', Admiral Owen Paris has the rank of Full Admiral. In season 7 episode 6 'Inside Man', the same Character is seen as being only a Vice Admiral. In the season 7 season episode 'Endgame', the last episode of the show, he is once again seen as being a Full Admiral.
Plot hole: Tuvok says that he was born on Stardate 38774. But he was born in 2264, when Stardates were only counted in the thousands, not the tens of thousands. If he were really born on SD 38774, that would mean he was born only a few years before the Enterprise-D was launched.
Imperfection - S7-E2
Continuity mistake: Tom Paris is already wearing his wedding ring in the scene where he is assisting the Doctor with Sevens operation. He doesn't get married until the next episode.
Repression - S7-E4
Other mistake: In the 3D cinema scenes in the Holodeck, Tom Paris and B'Elanna Torres can be seen using blue/cyan anaglyph glasses and the movie screen can be seen to be in blue/cyan also. This is fine and to be expected. However the shot in which the projection room can be seen behind the couple show the projection is through magenta/green filters. (00:03:30)
Repression - S7-E4
Continuity mistake: In the theater, the assailant's hand goes from a few inches away from the Tabor's position, to touching Tabor's position.
Continuity mistake: When Seven takes the mobile emitter off, it alternates position a few times. First, it's right side up. Then upside down in the close up, then right side up when the Doctor appears.
Visible crew/equipment: When the camera turns to face Tuvok as he approaches the crazed Hirogen, two stage crew members can be seen at the left side of the screen. The first is a rather plump man wearing what appears to be dark pants, and a short sleeve light shirt, while the second is squatting down in front of the first wearing shorts. (00:09:05)
Visible crew/equipment: After Paris hails Tuvok about the problem in the mess hall, a brawl breaks out. When the new Hirogen Alpha enters the galley to access the com relay, just as this Alpha removes the front panel in order to transmit a signal, the actor's green tape mark is visible on the floor. (00:39:00)
Other mistake: On the planet, as the camera looks at the Harojan from behind, the zipper for his costume is visible.
Plot hole: It's stated that only objects or people infused with the chronoton serum can pass through the time barriers, which is why the equipment belts and hyposprays had to be specially modified (these wouldn't be needed if the chronotons were producing a protective field wider than the infused object). But some un-infused artificial objects, such as clothing and com-badges, can pass through them without vanishing. This is most obviously seen in the scene when Chakotay pulls the captain through the barrier, since he doesn't even wait long enough for the serum to spread throughout her entire body, yet alone to perspire into her uniform.
Continuity mistake: When Neelix shows Torres her own quarters, her bat'leth can be seen in the background. It's a run of the mill bat'leth, unlike the one she actually hung up in 'Prophecy', which was gunmetal and had a spike in the middle.
Human Error - S7-E18
Audio problem: When Chakotay comes into Seven's quarters at some stage, the comm badge sound is used, not the standard door chime sound.
Visible crew/equipment: During B'Elanna's turn as the EMH protagonist in the Doctor's holonovel, when mustached "Marseilles" (Tom) enters the room, there are two red T-marks on the floor near the door. Then during Neelix's version as the EMH, when "Captain Jenkins" (Janeway) orders "Tulak" (Tuvok) and "Kymble" (Kim) to take the EMH for reprogramming, white T-marks are visible on the floor. (00:11:40 - 00:15:15)
Answer: If you include the original Star Trek series (1966) then there are several. The communicators used in the original series were before (and said to inspire) mobile phones. We currently do have teleportation technology but it currently only works on things the size of a few molecules. A "Cloaking device" also exists; it's a fabric that bends light through it, though it currently only works in infra-red. The Hypospray is real and was patented in 1960 - six years before the original series aired - it's actually called the Jet Injector. Faster Than Light travel is still a few decades off, but there are several real-world theories that look promising, including one that is remarkably similar to the method used in the Star Trek Universe called the Alcubeierre Drive that involves manipulating spacetime ahead and behind the ship and the ship "riding" it. Medical techniques and technologies have also advanced considerably; prosthetics particularity and we routinely have robots performing surgeries where absolute precision is needed. The "Shield" used in the series have a few primitive versions around. The Phasers used in the series are used but are not very powerful (nor will they ever be as powerful as the Star Trek version the laws of physics gets in the way) but rail-guns (using magnets to spin then propel a projectile) and particle accelerators like the Large Hadron Collider have been around for a while. The Replicator would require a nuclear fusion reactor and a nuclear fission reactor in something the size of a large oven and the Holo-deck wouldn't work at all based on our current understanding of physics so those are both still science fiction at the moment, but who knows!
Sanguis