We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧$\Delta \left(\right|\Delta u_i{|}^{p-2}\Delta u_i)=\lambda F_{u_i}(x,u₁,\dots ,uₙ)$ in Ω, ⎨ ⎩$u_i=\Delta u_i=0$ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.