An asymptotic solution to the Giesekus constitutive model of polymeric fluids under homogenous, oscillatory simple shear flow at large Weissenberg number, Wi >> 1, and large strain amplitude, Wi/De >> 1, is constructed. Here, Wi = lambda*gamma(0)*, where lambda* is the polymer relaxation time and yo is the shear rate amplitude, and De = co* is a Deborah number, where e is the oscillation frequency. Under these conditions, we show that the first normal stress difference is the dominant rheological signal, scaling as G*Wi(1/2), where G* is the elastic modulus. The shear stress and second normal stress difference are of order G*. The polymer stress exhibits pronounced nonlinear oscillations, which are partitioned into two temporal regions: (i) A "core region," on the time scale of r, reflecting a balance between anisotropic drag and orientation of polymers in the strong flow; and (ii) a "turning region," centered around times when the shear flow reverses, whose duration is on the hybrid time scale (lambda*)(2/3) (gamma(0)*)(1/3), in which flow -driven orientation, anisotropic drag, and relaxation are all leading order effects. Our asymptotic solution is verified against numerical computations, and a qualitative comparison with relevant experimental observations is presented. Our results can, in principle, be employed to extract the nonlinearity (anisotropic drag) parameter, a, of the Giesekus model from experimental data, without the need to fit the stress waveform over a complete oscillation cycle. Finally, we discuss our findings in relation to recent work on shear banding in oscillatory flows. (C) 2016 The Society of Rheology.