Plot hole: The entire premise of the episode is contradictory; the franchise makes abundantly clear through the so called Prime Directive that the Federation abides to a code (often creating moral dilemmas that may require to stretch the rules) that says that their staff is not supposed to interfere in world that haven't reached warp capabilities, nor involve themselves in their internal matters. In the first half of the episode, Riker and the others just go 'mingle' with the frisky natives without a care in the world, and yet in the second half the Prime Directive itself is referenced explicitly and it is part of the plot.
Plot hole: The binars needed two people to upload the data back to their world but they only intended on keeping Riker in the holodeck. Riker is just one person. Picard came into the holodeck purely by accident - even Minuet said this was true. If Picard hadn't been there, Riker could not have uploaded the data on his own and the binars' plan would have failed.
Plot hole: When the image of the warp bubble closing in on the superimposed "Enterprise" image, Dr. Crusher asked for it to be the same scale for each image. At the rate the bubble was closing in on the engineering section, it would have been destroyed LONG before the 4 minutes 15 seconds the computer gave for it to be destroyed.
Plot hole: About two thirds of the way through this episode, Data deduces how Moriarty was able to 'leave' the holodeck. The big reveal is that he never did leave the holodeck, he merely reprogrammed it to simulate the rest of the ship without Picard, Barclay or Data's knowledge. While it is believable that this would fool humans like Picard and Barclay, it is ludicrous to suggest that Data would be taken in by it, even for a second. Data is an android whose perceptions of sight, sound and the world around him are far more sophisticated than humans. To list what we know of Data's perceptive abilities from previous episodes would take up the entire page, but suffice it to say he should have immediately recognised the 'Enterprise' as force fields and holograms rather than the genuine article. Note: this goes beyond a 'character mistake' or anything like that. Data's enthusiasm for Sherlock Holmes style deduction should have led him to eliminate the impossible before considering what was probable. Data accepts the impossibility of a holodeck character existing is the real world before discounting the possibility that it was still a holodeck simulation. This contradicts an awful lot of what is known about Data's abilities and powers of deduction. Although it serves for a good mystery the fact is it takes Data far longer than it should to deduce, or even guess at, the truth of their situation.
Plot hole: During a private conversation with Worf, (the not yet revealed to be clone of) Kahless recounts a time during Worf's childhood when the actual Kahless appeared to Worf in a vision and told him he would do something no other Klingon had ever done. However, it would be impossible for a clone to have such a memory, as his creators would have no knowledge that such a memory even existed, let alone the circumstances and specific content of that vision.
Plot hole: When Picard, Geordi, and Troi encounter one of the first time disturbances it results in one of the runabout's nacelles using up all of its fuel. Data says this is due to it having been in operation for 47 days (according to the plasma conversion sensor). After this the Captain reaches for the bowl of rotting fruit which causes him to scream in pain. The other crew members rush in and Troi scans his hand. She tells Picard that his hand is metabolizing at approximately 50 times the normal rate. Data and Geordi then discover that the temporal disturbance which covers the fruit also covers the nacelle that has lost all of its fuel. Data also notes the disturbance extends outward from the hull, about 17 meters from the ship and is spherical in shape. The problem here is that according to what Data said earlier the engine was active for 47 days even though it had only very briefly come into contact with the time distortion - maybe 1 second at most, but likely far less time than this since the ship was at warp when the fuel was consumed. So if we assume the engine was in contact with the fragment for 1 second then time is actually moving at 4,060,800 times the normal rate - not 47 or 50 times normal. (00:11:00 - 00:12:00)
Plot hole: They have already shown that anything that comes into contact with the time bubbles is immediately affeceted. The engine running 47 days non-stop, the fruit againg, Picard's hand aging, etc. Geordi creates subspace isolation field that allows them to beam off the shuttle and onto the other ships to investigate. Problem is, the second the transporter beam hit the time bubble, it would have stopped and never materialized because it would have been affected by the extremely slow passage of time in that bubble. The warp core breach was affected as was the disruptor fire so the transport beam would have been easily affected. And they have never shown that have been able to alter the transporters in such a manner that would allow them to beam over.