What would happen if you caught a PokeBall with another PokeBall? Consider the following scenario: A trainer is trying to catch a rare or strong Pokemon, and figures that catching it in a PokeBall will have a low likelihood of actually catching and containing it. 

What would happen if, after the trainer threw the first PokeBall and before the wild Pokemon breaks free, the trainer hit the occupied PokeBall with a second PokeBall? Would the both the original PokeBall and the wild Pokemon be "caught" by the second PokeBall? Would the wild Pokemon just be transferred from the first PokeBall to the second PokeBall? Would this double the likelihood that the Pokemon will be caught, due to the fact that it has to break out of two balls instead of one? 

If the PokeBalls can be "layered," and the probability that you catch a Pokemon increases with how many PokeBalls you can layer on it, could you not just keep layering PokeBalls until you reach the strength of a Great or Ultra Ball?

We already know that inanimate objects and things other than Pokemon can be "caught" by PokeBalls, as evidenced by this question.
While it's not an ordinary Pokéball, the Mewtwo ball is capable of capturing another Pokéball inside it.



This is demonstrated in the first Pokemon movie and quoted in the synopsis section of the movie's wiki page. A gif of the event is below too:


  "Ash tries returning Bulbasaur and Squirtle (who have been fighting the balls bravely), but it's no use. Two of Mewtwo's Poké Balls fly over and capture Bulbasaur and Squirtle's Poké Balls."




Granted, this is a special kind of ball developed by Mewtwo, so I can't say for sure in this answer whether a standard Pokéball would behave similarly.

Additionally, the catch rate for the Mewtwo ball is listed as 100% on the wiki page. Therefore it's impossible to say whether it would increase the catch chance of a wild pokémon through stacking, as it's a guaranteed capture with just one.