This paper defines a hierarchical version of the maximum flow problem. In this model, the capacities increase over time and the resulting solution is a sequence of flows that build on each other incrementally. Thus far, hierarchical problems considered in the literature have been built on NP-complete problems. To the best of our knowledge, our results are the first to find a polynomial time problem whose hierarchical version is NP-complete. We present approximation algorithms and hardness results for many versions of this problem, and comment on the relation to multicommodity flow.