How many games of Magic: The Gathering can be played?

Question

The number of possible decks is incomprehensibly large. See this question for context.

However, the number of games that can be played is incomprehensibly larger—for every deck you have, there are about 60! ways to draw the cards in order (depending on the size of your deck). This doesn’t even take into account the way that you could die before drawing all of those cards, or the number of actions you can take during each round (or not take).

Given that every single card presents new possibilities to the set of possibilities from previous cards, is it even practical to try to calculate this?

I am here defining a game as a discrete set of actions that could be taken by two players while properly following the rules of Magic: The Gathering, and the sequence of collections of cards included in each players’ respective draw piles, graveyards, hands, and exile piles over time (for those familiar with the terminology). (Technically an unlimited number of players could play Magic: The Gathering against each other in a single game, but to avoid the trivial case of an infinite number of games being played, I’m restricting the rules to the typical case of two.)

Answer 1

Uncountably infinite

1. Both players each play 6 lands and a Leashling.
2. For any given real number between 0 and 1, consider its binary representation.
3. During each player's turn, that player returns their Leashling to their hand, returns one of two cards in their hand to the top of their library depending on the next binary digit, and then re-plays the Leashling. They then draw back that card at the beginning of their next turn.