When Josh and his boss are playing piano at the toy store, Josh throws a bag down on the ground before playing. When Josh and his boss leave Josh leaves the bag on the ground and doesn't take it with him.
When Billy's mother is complaining at the dinner table, Billy goes to remove a roast from the oven. When he takes it out of the oven using a pot holder, he also grabs it with his other hand which would result in him being badly burned.
When Josh is telling Susan about his electronic comic idea, the sleeve on her left shoulder keeps changing. One shot it's off the shoulder, then it's back on the shoulder, then it's off the shoulder again. This happens several times.
When Josh and MacMillan are on the giant piano, the crowd behind them changes a lot between shots, e.g. a young girl in the background is seen to sit on her dad's lap twice. Also, the boy in the sweater disappears, then reappears in time to applaud at the end of the song.
The story seems to start in late October/early November and this is confirmed when both Billy accompanies Josh to a "birthday" party and his mother later mentions to Billy "...he had a birthday...". Given his "missing" information on the milk carton puts his birthday on January 20th, and tracing the time-line backwards with the company party that seems to be around Christmas, and the clerk giving a date of "6 weeks, maybe longer" for the location of the fair reports to become available, then the story probably started around Thanksgiving, went through Christmas, and ended sometime in late January/early November when Josh got his second wish... Yet, the foliage on the trees in New York is green and/or bright yellow indicating summer and fall.
When young Josh is standing in front of the Zoltar machine for the first time, Zoltar is opening and closing its mouth and its eyes are flashing red, but when we see it from the side, the head is still.
When Susan chases after Josh after he runs out of the meeting, she just misses the elevator doors closing. She pushes the down button to call another elevator, but it was already lit - there's no reason for it to be lit when a descending elevator has just closed and left and nobody else is there to push the button.
When Josh is discussing his electronic comic idea with Susan, the slice of pizza in the middle of the table has its crust facing the camera in the first shot, then it moves 90 degrees around the plate.
(at around 37 mins) After Josh is "shot" by the toy laser gun in FAO Schwarz, he falls down to the floor quivering. The black laser power box is hanging down on the right side of his chest, with Josh on his back. When Robert Loggia asks "You work for me, don't you", the black power box is now hanging down on the left side of Josh's chest.
The fair at the beginning of the movie was set in New Jersey (the George Washington Bridge was visible). Josh and Billy filed an information request with the New York City Department of Consumer Affairs, which wouldn't cover New Jersey. The would have needed to go to the equivalent office for Bergen County, New Jersey.
The morning after Josh gets "big", he tries to convince his mom that he is really her son. He tells her that his birthday is November 3. Later, we see a milk carton with a "missing" advert on the side about Josh, which says his birthday is 1st January.
When 30 year old Josh is on his job interview, the interviewer tells him that he "seemed to have forgotten a few numbers on his social security number." Josh apparently does not know his SSN and even makes up the last few numbers, which the interviewer writes down. If his SSN was indeed incorrect, then the payroll department of FAO Schwartz would have not been able to process any taxes from his paycheck, thereby he would have not received a paycheck when they were handed out.