u/Chemical_Ad_4073

We want to find out how fast this down arrow grows. To assess that efficiently, we focus on one thing at a time.

One arrow (a↓b):

This is plain old exponentiation. 2↓2=4, 2↓3=8, 2↓4=16, 3↓2=9, 4↓2=16, 3↓3=27, 4↓4=256.

Two arrows (a↓↓b):

There is where it gets interesting. This is weak tetration. Here are some values:

1↓↓x from 1 to 5: 1, 1, 1, 1, 1

2↓↓x from 1 to 5: 2, 4, 16, 256, 65536

3↓↓x from 1 to 5: 3, 27, 19683, 7.626*10^12, 4.434*10^38

4↓↓x from 1 to 5: 4, 256, 4.295*10^9, 3.403*10^38, 1.341*10^154

5↓↓x from 1 to 5: 5, 3125, 2.98*10^17, 2.351*10^87, 7.182*10^436

Three arrows (a↓↓↓b):

This one is the biggest of these. This is weak pentation.

a↓↓↓b	b=1	b=2	b=3	b=4
a=1	1	1	1	1
a=2	2	4	256	3.232*10^(616)
a=3	3	19683	10^(1.664*10^9)	10^(103.327*10^9)
a=4	4	3.402*10^(38)	10^(1.518*10^117)	10^(104.555*10^117)