By applying polymer scaling theory, we investigate the effects of steric interactions on the growth of a class of equilibrium polymers that require activation before polymerization can take place. The growth exponents that we obtain in dilute and semidilute solutions of such thermally activated supramolecular polymers are identical to those for isodesmic equilibrium polymers, but different from those of "living" supramolecular polymers that are chemically activated by reaction with an initiator. We also find that with increasing activation energy the crossover from the dilute to the semidilute regime shifts toward the polymerization transition and eventually merges with it, giving rise to nonclassical behavior in particular of the osmotic compressibility. It appears that steric interactions make the polymerization more co-operative in dilute solution, but less so in semidilute solution.