Optimal resource allocation (RA) in massive carrier aggregation scenarios is a challenging combinatorial optimization problem whose dimension is proportional to the number of users, component carriers (CCs), and OFDMA resource blocks per CC. Towards scalable, near-optimal RA in massive CA settings, an iterative RA algorithm is proposed for joint assignment of CCs and OFDMA resource blocks to users. The algorithm is based on the principle of successive geometric programming approximations and has a complexity that scales only linearly with the problem dimension. Although its derivation is based on a relaxed formulation of the RA problem, the algorithm is shown to converge to integer-valued RA variables with probability 1 under mild assumptions on the distribution of user utilities. Simulations demonstrate improved performance of the proposed algorithm compared to commonly considered heuristic RA procedures of comparable complexity.