A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the reals which are computable from the halting problem. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real we assign a degree structure, the structure of all possible ways available to approximate the real. So the main criterion in this classification is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results.