You can always combine integer operations in smaller chunks to simulate something that's too big to fit in a register. Python even does this transparently for you, so your integers can be as big as you want.

The fundamental problem that led to requiring 64-bit was when we needed to start addressing more than 4 GB of RAM. It's kind of similar to the problem of the Internet, where 4 billion unique IP addresses falls rather short of what we need. IPv6 has a host of improvements, but the massively improved address space is what gets talked about the most since that's what is desperately needed.

Going back to RAM though, it's sort of interesting that at the lowest levels of accessing memory, it is done in chunks that are larger than 8 bits, and that's been the case for a long time now. CPUs have to provide the illusion that an 8-bit byte is the smallest addressible unit of memory since software would break badly were this not the case, but it's somewhat amusing to me that we still shouldn't really need more than 32 bits to address RAM at the lowest levels even with the 16 GB I have in my laptop right now. I've worked with 32-bit microcontrollers where the byte size is > 8 bits, and yeah, you can have plenty of addressible memory in there if you wanted.