Computers also go arbitrarily large. Not infinite, but arbitrarily large.

A real number type could be bounded by the amount of RAM you have.

It is possible within the limits of available memory on the computer. Rather than the usual limit of a fixed number of bits.

I've ironically found that big integer libraries sometimes optimize math routines more than string conversion. This was quite annoying for me when I optimized the factorial function in Ruby, and found that generating the string was my bottleneck. I then optimized that as well. :-)

They can go large enough for anything that matters.

A quick Google says there's an estimate of 10^78 to 10^82 atoms in the universe. That number would be able to be stored in well under 300 bits.