I think any number whose information density is more than could be memorized becomes data, and no longer counts as a real life number.

One might stretch the definition to allow numbers that you could read on a postcard in one go to still count as numbers, but it’s a big stretch. If there’s a more understandable representation of the number — especially if there’s a more understandable representation — then it’s data and not a number.

Beyond a certain amount of information content numbers just become data. Certainly at the point where counting the number of digits is non trivial. (There are notations to make magnitude easier to see, but these purposefully delete information content.)