We consider a sequence (Z_n)_n≥1 defined by a general multivariate stochastic approximation algorithm and assume that (Z_n) converges to a solution z* almost surely. We establish the compact law of the iterated logarithm for Z_n by proving that, with probability one, the limit set of the sequence (Z_n - z*) suitably normalized is an ellipsoid. We also give the law of the iterated logarithm for the l^p norms, p ∈ [1, ∞], of (Z_n - z*).