$\infty$ is not the answer I'm looking for.
Let's say I had a stupidly big number in my head and I wanted to write it down on a sheet of paper. I could start plotting down the digits, but at some point I would run out of space to write. So I could use more sheets of paper, but again, at some point I will run out of paper. I could try to write it in scientific notation, but then again, at some point I will run out of space to write the exponent. Even if I tried to chain together functions that grow very fast ( like $y = f( f( f(\ldots)))$.
where $f(x)$ grows very fast ), at some point I will run out of space by writing chains of functions. All of those techniques require more and more space as the number gets bigger, so is there a notation that lets me write an infinitely large number with finite characters, regardless of precision?
I'm asking because we often think of computers as the way to handle big number problems, but just like a sheet of paper, a bigger number requires more physical space in memory. So physical devices being finite in size, they will always have a limit on the size of the numbers they can represent with the standard notation systems. This is where ∞ doesn't work, since you can't manipulate it like a number and even if you could, there would be no difference between $\infty$ and $\infty+1$, therefore being useless to write down a specific number.
I have found this similar question, but the answer was talking about full-precision numbers, whereas I am only interested in the size of it, regardless of precision.
It's impossible to encode real numbers with finite strings regardless of how simple or complex the algorithm is. That's because the space of finite strings is countable while the reals are uncountable. Numbers that can represented with finite strings are called computable numbers. Due to the aforementioned uncountability almost all real numbers are uncomputable. The computable numbers do form a field so computations with them matches our intuition.