The critical amplitude of the q-state Potts-model free energy is studied as a function of q in two dimensions and on the diamond hierarchical lattice. The amplitude diverges at an infinite number of q values, qn,introducing logarithmic terms in the free energy. We expect that in each interval (qn,qn+1) there is a q value where the amplitude vanishes, affecting the singularity of the free energy as a function of temperature. Possible consequences for gelation and vulcanization of polymers are discussed.