A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v₁v₂…v_2l for which c(v_i)=c(v_l+i) for all 1≤i≤l. Given graphs G and H with |V(H)|=k, the lexicographic product G[H] is the graph obtained by substituting every vertex of G by a copy of H, and every edge of G by a copy of K_k,k. We prove that for a sufficiently long path P, a nonrepetitive coloring of P[K_k] needs at least 3k+⌊k/2⌋ colors. If k>2 then we need exactly 2k+1 colors to nonrepetitively color P[E_k], where E_k is the empty graph on k vertices. If we further require that every copy of E_k be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P[E_k] is at least 3k+1 and at most 3k+⌈k/2⌉. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.