There has to be some procedure how to come up with "new" numbers though, if you want to have more in the end than just a fancy binary tree - in particular if you want to map your "fake numbers" to the reals, infinity, etc.

This procedure is enough. If you define addition and other operations in a certain way (as Conway did), it turns out that on the omega-th day (i.e. after initial infinite steps), all reals will be born.