Let (nk)k≥1 be a lacunary sequence, i.e., a sequence of positive integers satisfying the Hadamard gap condition nk+1/nk≥q>1, k≥1. By a classical result of Philipp (Acta Arith. 26 (1975) 241–251), the discrepancy DN of (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm, i.e., we have 1/(4√2)≤lim supN→∞NDN(nkx)(2N log log N)−1/2≤Cq for almost all x∈(0, 1), where Cq is a constant depending on q. Recently, Fukuyama computed the exact value of the lim sup for nk=θk, where θ>1, not necessarily an integer, and the author showed that for a large class of lacunary sequences the value of the lim sup is the same as in the case of i.i.d. random variables. In the sublacunary case, the situation is much more complicated. Using methods of Berkes, Philipp and Tichy, we prove an exact law of the iterated logarithm for a large class of sublacunary growing sequences (nk)k≥1, characterized in terms of the number of solutions of certain Diophantine equations, and show that the value of the lim sup is the same as in the case of i.i.d. random variables.