We use high precision lattice simulations to calculate the plaquette expectation value in three-dimensional SU(N_c) gauge theory for N_c = 2,3,4,5,8. Using these results, we study the N_c-dependence of the first non-perturbative coefficient in the weak-coupling expansion of hot QCD. We demonstrate that, in the limit of large N_c, the functional form of the plaquette expectation value with ultraviolet divergences subtracted is 15.9(2)−44(2)/N_c².