We present a scaling theory of the living polymerization of material in a good solvent for the case where the total number of chains is fixed by an irreversible initiation reaction and conclude that excluded-volume interactions have only a very weak influence on the growth of the polymers. Our estimate for the crossover temperature to the semidilute regime is in good agreement with more elaborate calculations based on a mapping onto a magnetic problem and with experimental data. We find that the polymerization temperature and the crossover temperature to the semidilute regime merge in the limit of vanishing initiator concentration and discuss the implications this has for the nonclassical behavior of living polymers in the vicinity of this limit.