Is there a (more or less complete) theory of real numbers, where every number except for $1$ is defined by a combination of arithmetic operations acting on $1$.
I know we can build any whole number by using addition and subtraction of 1s, and any rational number by using division of whole numbers.
We can also build any irrational number by using infinite series with rational terms or infinite products or infinite continued fractions.
So it is not a question about whether it is possible to build all real numbers that way, but rather is where some treatise on the subject?
There are numerous constructions of the real numbers, some of which may be along the lines of what you are looking for (do note your question in not entirely clear). You may be interested in the survey article here (or its arXiv version) which surveys many constructions.