You can get many many more than 256 with subnet masks.

Each of the original 256 addresses has its own connection to the main trunk, behind that one address, are smaller trunks that all use same process and it can be repeated over and over to tiny little leafs.

Every DNS server has a SOA, Start of Authority, just a pointer, pointing upstream to another up stream pointer. And now with dynamic IP address assignments you can have way more machines connected because they "give back" their address when not in use.

There is a finite number tho. That's one reason there are .com .edu etc.

Managed correctly there are more addresses available for large companies to ever run out.