A Flory-Huggins type lattice model of living polymerization is formulated, incorporating chain stiffness, variable initiator concentration r, and a polymer-solvent interaction chi. Basic equilibrium properties [average chain length L, average fraction of associated monomers Phi, specific heat C-P, entropy S, polymerization temperature T-p, and the chain length distribution p(N)] are calculated within mean-field theory. Our illustrative calculations are restricted to systems that polymerize upon cooling [e.g., poly(alpha-methylstyrene)], but the formalism also applies to polymerization upon heating (e.g., sulfur, actin). Emphasis is given to living polymer solutions having a finite r in order to compare theory with recent experiments by Greer and co-workers, whereas previous studies primarily focused on the r --> 0(+) limit where the polymerization transition has been described as a second order phase transition. We find qualitative changes in the properties of living polymer solutions for nonzero r: (1) L becomes independent of initial monomer composition phi(m)(0) and temperature T at low temperatures [L(T much less than T-p)similar to 2/r], instead of growing without bound; (2) the exponent describing the dependence of L on phi(m)(0) changes by a factor of 2 from the r --> 0(+) value at higher temperatures (T greater than or equal to T-p); (3) the order parametertype variable Phi develops a long tail with an inflection point at T-p; (4) the specific heat maximum C-P(*) at T-p becomes significantly diminished and the temperature range of the polymer transition becomes broad even for small r [r similar to O(10(-3))]. Moreover, there are three characteristic temperatures for r > 0 rather than one for r --> 0: a "crossover temperature" T-x demarking the onset of polymerization, an r-dependent polymerization temperature T-p defined by the maximum in C-P (or equivalently, the inflection point of Phi), and a "saturation temperature" T-s at which the entropy S of the living polymer solution saturates to a low temperature value as in glass-forming liquids. A measure of the "strength" of the polymerization transition is introduced to quantify the "rounding" of the phase transition due to nonzero r. Many properties of living polymer solutions should be generally representative of associating polymer systems (thermally reversible gels, colloidal gels, micelles), and we compare our results to other systems that self-assemble at equilibrium.