I've watched the Back to the Future series countless times and if you're going to make a movie about time travel, there's bound to be some mistakes/paradoxes. This paradox is one that probably exists but has probably not been brought up properly before. It's going to be about the second film in the franchise, Back to the Future Part II.
SPOILERS. Duh. And you can skip the summary if you want.
In Back to the Future Part II, Doc brings Marty and Jennifer to October 21, 2015 due to a problem with their future children.
Doc knocks out Jennifer with a sleep inducing alpha rhythm generator (which still doesn't exist) and Marty successfully saves their future children from decades of imprisonment. However, while Marty is in 2015, he buys a book called Gray's Sports Almanac, which covered the results of many sporting events from 1950-2000.
He planned to use it once he went back to 1985 so that he could bet on games and become rich. But when Doc finds out, he throws the book away for reasons I can never understand. A 77-year old Biff knows Marty's plan and steals his idea. Jennifer is bought to her future home by two police officers that thought she was the Jennifer of 2015. Marty and Doc go and save her, thinking it's a good idea to leave a DeLorean time machine out in the open with the keys in the engine and the door open.
Old Biff steals the time machine with the almanac in hand and goes back to November 12, 1955, which is actually the same day that Marty traveled from 1955 to 1985 in the original Back to the Future. He brings the time machine back without Marty or Doc knowing.
So Marty and Doc go back to 1985, but when they go back, they discover that it's a different 1985. A dystopian 1985.
Hill Valley was now Hell Valley. Marty finds out what Biff did and everything that happened. Marty finds out Biff had used the almanac to become Donald Trump, basically. He also finds out that in 1973, when he was five years old, Biff shot and killed George to marry Lorraine. The Marty of this 1985A was in Switzerland attending a boarding school and the Doc of this 1985A had been committed to a mental asylum.
But this was 1985 as everyone knew it. Everyone except Marty, Doc, and Einstein.
Wait...what? What about the other Marty?
As in this Marty.
The Marty from the first movie.
This Marty, from the first movie, travels from 1955 to 1985 the same day Biff gets his hands on the sports almanac.
But he should have gone back to the 1985 that Biff altered. He should have gone to the 1985A, where Biff is successful and Marty's dad, George is dead, and Lorraine is an alcoholic.
So technically, there should be two Martys in 1985A, wandering around wondering what the hell happened.
That other Marty, the one that goes to 1985 in the first movie, should also have gone back to his old house.
So shouldn't there be two Martys? One that just came from 1955 and one that just came from 2015? And this would be a paradox because if Marty ends up traveling to 1985A, how does he end up going to 2015 to screw things up anyway? And how should there even BE a time machine if Doc had been committed years before?
If you want to read more Back to the Future paradoxes, click this.