Moving contact line flows subject the fluid to infinitely growing deformation rates as the distance to the contact line decreases. When a fluid whose constitutive relation depends on a characteristic time scale enters the region where the inverse deformation rates are comparable to its time scale, the fluid behavior will exhibit non-Newtonian behavior impacting both the stresses and, for a system with deformable interface, the flow kinematics. We will discuss measurements of spreading dynamics of various fluids: from a "simple" very flexible polymer with Newtonian rheology, to a slightly non-Newtonian oligomer, to a highly elastic Boger fluid. All these fluids show deviations from a model that assumes Newtonian behavior of the fluid throughout the flow near the contact line. Our results suggest that the wetting behavior of all these polymer melts, even those nominally Newtonian in rheological measurements, are significantly influenced by non-Newtonian behavior in some region of the wedge flow near the moving contact line. This influence is present even when all modes in the fluid are fully relaxed over much the wedge flow near the contact line.