Why is $2^{2^2}$ so much less than $2^{2^{2^2}}$?

Question

I assume there's a name for raising something by an exponent repeatedly, but I haven't been able to find it. I understand why $2^{2^2} = 16$ and $2^{2^{2^2}} = 65536$ by plugging in the numbers, but I am having trouble building some intuition for why the former is so much smaller than the latter; $2^{2^{2^{2^2}}}$ can't even be handled by my calculator, for instance.

Answer 1

I assume there's a name for raising something by an exponent repeatedly, but I haven't been able to find it.

There is indeed a term for that; it's called tetration

It might be easier to wrap your head around the concept if you use 10 instead of 2.

$10^{10}$ is a 1 with ten zeroes after it. Go ahead and write that on a piece of scrap paper real quick. (it equals ten billion if you're wondering)

$10^{10^{10}}$ is a 1 with ten billion zeroes after it. If you wrote one digit every second, it would take you 300 years to finish writing!

Answer 2

The name you are looking for is "tetration" or "hyper-$4$". It is a hyper-operator, indeed. Tetration is the operator such that "addition : multiplication = multiplication : exponentiation = exponentiation : tetration", even if we lose some properties climbing the aforementioned ladder (e.g., $2 + 3 = 3 + 2$, $2 \cdot 3 = 3 \cdot 2$, while $2^3 \neq 3^2$, and also $^{2}3 \neq ^{3}2$, since $2^{\left(2^2\right)} \neq 3^3$).

Now, you can think that we are calculating those giant numbers as $^{5}2$ solving the power tower from top to bottom, so that $^{4}2$ becomes the exponent of $^{5}2$ and $2^{65536}$ is about $2.003529930 \cdot 10^{19728}$, which is a number with almost $4000$ times the digits of $^{4}2$.

Now, we can understand why $^{6}2$ is as big as $10^{{6.031226063} \cdot 10^{19727}}$.