The monthly crew pairing problem (CPP) consists of determining a least-cost set of feasible crew pairings (sequences of flights starting and ending at a crew base) such that each flight is covered once and side constraints are satisfied. This problem has been widely studied but most works have tackled daily or weekly CPP instances with up to 3500 flights. Only a few papers have addressed monthly instances with up to 14000 flights. In this paper, we propose an effective algorithm for solving very large-scale CPP instances. This algorithm combines, among others, column generation~(CG) with dynamic contraint aggregation (DCA) that can efficiently exploit the CG master problem degeneracy. When embedded in a rolling-horizon (RH) procedure, DCA allows to consider wider time windows in RH and yields better solutions. Our computational results show, first, the potential gains that can be obtained by using wider time windows and, second, the very good performance of the proposed algorithm when compared to a standard CG/RH algorithm for solving an industrial monthly CPP instance with 46588 flights.