A lot of articles seems to be concerned with "unimaginably" large numbers, leading to a conclusion that we just do not have sufficiently good notation for dealing with such numbers.
So imagine we have a discrete Ackermann sequence based function that has values of:
A(1) = 1
A(2) = 2+2 = 4
A(3) = 3*3*3 = 27
A(4) = ⁴4 ≈ 10 ^ (10 ^ 154)
How can we compute the value of A(3.5)? Let's call it 3.5⇑. The function should be defined over reals ≥ 1 and have all of its n-th derivatives smooth. The approach should produce a practically usable function body for small parameter values even though it may become unbound as parameter grows.
If we use the N-ation of N notation, we can get rid of hype of large numbers in math not unlike how scientific notation allowed getting rid of too large or too small numbers in physics.
Most of commonly used numbers will fit between 3⇑ and 4⇑.
Predicated on that we can calculate what's 3.2⇑ + 3.4⇑.
The fact that we cannot "write down" large numbers⇑ such as 15⇑ should not discourage us - math operations may still be carried out on them. After all, it's not like physicists decry the impracticality of writing down Avogadro number in simple decimal notation.