We study approximation properties of centred additive random fields Y _d , d∈N. The average case approximation complexity n ^{Y _d} (ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Y _d , with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of n ^{Y _d} (ε) for arbitrary fixed ε∈(0,1) and d→∞. Under natural assumptions we obtain general results concerning asymptotics of n ^{Y _d} (ε). We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.