Anyway, ,

So if quantum computers manages to utilise the atom, as a bit
recording device, where 1 integer can be stored using a single
atom . . .The total number of integers that can be used for any
number, even if in the most effient form such as. . .

9^9^9^9^9^9^9^9^9^9^9^9^9^9 etc etc. . .

Which is approx

1.59486 × 10^(55) kg / 1.660538782(83)×10-27 kg

After that, I agree that we could resort to other symbols which gave
meaning to numbers.

for example

K^K^K^K^K^K^K^K^K^K^K^K^K^K etc etc. . .

However, once you run out of numbers, you run out of numbers of
symbols. Beyond this, whetever it was that was recording will have
lost mathematical meaning.