Forum:Fast-growing function came up in a game

Fooling around with Magic: The Gathering cards, I accidentally came across something of googological interest.

I will try and explain things in terms for people who don't know the game well, but for those interested, the relevant cards are Doubling Season, Clone Legion, and Opalescence.

We start with 4 cards out, they all have this effect: "When you make a copy of one of your cards, double the number of copies made."

This effect stacks geometrically, so having 4 copies means we get a multiplier of 2^4 = 16

We also have an ability, "Make a copy of every card in play."

Using this ability once, we produce 2^4 copies each for our 4 cards, for a total of 64, adding to our existing 4 for 68 cards in play.

Now we have 68 cards, bringing our multiplier to 2^68 = 295,147,905,179,352,825,856

The second time we use the ability, we make 2^68 copies each of our 68 cards, bringing us to a total 20,070,057,552,195,992,158,276 cards.

With this many cards, our multiplier becomes 2^20,070,057,552,195,992,158,276, which is number of about 6*10^21 digits.

The third time we use the ability, we produce approximately 10^(10^21.7) more cards. You see where this is going!

It is possible within the game rules to use this ability an arbitrary number of times, so we can produce some numbers that are infeasible to compute.

I guess our function is...

f(0) = 4

f(n) = f(n-1) * 2^f(n-1)

In the grand scheme of googology this is not an asoundingly powerful function, but it was interesting to find it in an unexpected place.