Can you describe every number as a handful of small recurring operations and some input?

Question

Let's say that I would like to describe every number from 1 to googleplex with a limited number of operations, but including recursion.

So for a random example, I want to describe: 340282366920938463463374607431768211146

I can describe that as the following:

input = 2

while_loops = 7

while (while_loops > 0):
    input = input * input
    while_loops = while_loops - 1

input = input - 310

print(input)

Is it possible to represent all integers between 1 and googleplex with similarly simple steps?

Is there a way to measure how complex the algorithms would need to get to be able to describe every number between those two?

Is there a way to quickly determine what those algorithms are?

What field of mathematics would study this question? Are there any big name papers?