Abstract. Let G be a graph. By W(G) we denote its Wiener index and by Lⁱ(G) we denote its i-iterated line graph. We study the ratio R_k(G)=W(L^k(G))/W(G). We prove that if k>=3 and T is a tree on n vertices, then R_k(T) is smallest when T is a path. We pose several conjectures related to R_k(G).