How to own the most cards possible?
There is no limit to the amount of cards in your deck. For a given number N, the simplest way to own at least N cards during a gaming of Magic is to start with N cards in your deck.

100.5. If a deck must contain at least a certain number of cards, that number is referred to as a minimum deck size. There is no maximum deck size for non-Commander decks.

Most cards are limited to four copies each, but basic lands and cards like Relentless Rats are not.
Also, the only cards which change the owner of other cards are either silver-bordered or referring to ante and are outright banned in any competitive format.