Let X (t, ω) be an additive random field for (t, ω) ∈ [0, 1]^d × Ω. We investigate the complexity of finite rank approximationX (t, ω) ≈ underover(∑, k = 1, n) ξ_k (ω) φ{symbol}_k (t) .The results are obtained in the asymptotic setting d → ∞ as suggested by Woźniakowski [Tractability and strong tractability of linear multivariate problems, J. Complexity 10 (1994) 96-128.]; [Tractability for multivariate problems for weighted spaces of functions, in: Approximation and Probability. Banach Center Publications, vol. 72, Warsaw, 2006, pp. 407-427.]. They provide quantitative version of the curse of dimensionality: we show that the number of terms in the series needed to obtain a given relative approximation error depends exponentially on d. More precisely, this dependence is of the form V^d, and we find the explosion coefficient V.