Wikipedia (https://en.wikipedia.org/wiki/Observable_universe#Matter_con...) thinks there are 10^80 atoms in the observable universe, which is approximately 2^240. If you only need to assign a unique ID to each atom, then you need a counter with approximately 240 bits.

But if you needed to store even one bit of information per atom (e.g. whether it is hydrogen or not), then don't you need 2^240 bits uncompressed?