There are numbers that are so large that there is no compact formula to represent them. Think of a number so large, that its number of digits is lo large, that the number of digits of its number of digits is so large... and it goes on and on -- you get the idea.
Sure, if you are able to define such a number, then add one, or even 0.5, and you get an even bigger number. But this is not the point. The issue is to come up with such massive numbers in the first place. The biggest ones known to men are produced by unusual recursions, and such recursions can be used to test how powerful a computer is, when dealing with recursive algorithms.
Let's start with a very simple recursive function that quickly produces phenomenally large numbers [...] This is known as the Ackermann function. The fact that the recursion always ends no matter what n and m are, is by itself surprising. Below is an extract of the steps needed to compute A(4,3).