The arrow notation designed by Knuth in 1976 suggests a new series of numbers, which we name the Ackermann numbers after the creator of a very similar function from 1928, the Ackermann function. The first few numbers are:
1^1, 2^^2, 3^^^3, 4^^^^4, ...
These increase frighteningly fast. At first, 1^1 is 1, 2^^2 = 2² = 4. However, 3^^^3 is a "tower" of threes, and the number of threes is 3 ^ (3 ^ 3) = 7625597484987. That's an enormous number.
However, we've only just begun.
4^^^^4 = 4^^^4^^^4^^^4 =
4^^4^^4^^4 ... ^^4^^4^^4. How many fours? This many:
4^^4^^4 ... ^^4^^4, where the number of fours here is:
That's a tower of fours in which the number of fours is another tower of fours, in which the number of fours is a tower of 4 fours, ie 4 ^ a 155-digit number! And we're done.
That's just number A4.
Using Conway and Guy's chained arrow notation, the nth Ackermann number is simply n -> n -> n. Not that that makes them any easier to compute.
Adapted from The Book of Numbers, Conway & Guy 1995, and a few other sources also.