We consider vector functions u : ℝⁿ ⊃ Ω → ℝ^N minimizing variational integrals of the form ∫_Ω G(∇u) dx with convex density G whose growth properties are described in terms of an N-function A : [0,∞) →[0,∞) with lim sup_t→∞ A(t)t^-2 < ∞. We then prove - under certain technical assumptions on G - full regularity of u provided that n = 2, and partial C¹-regularity in the case n ≥ 3. The main feature of the paper is that we do not require any power growth of G.