Recently, we explored the effects of weak fluid elasticity (El ≪ 1) on the stability of co- and counter-rotating Taylor–Couette (TC) flows [Dutcher and Muller, J. Rheol. 55(6), 1271–1295 (2011)], where accessible flow states were primarily governed by the dominant inertial forces, yet modified by the weaker elastic forces. Here, the study of the inertial–elastic TC problem is expanded to El near unity, illuminating the effects of increasing the elastic forces on the inertially driven instabilities. A polyethylene oxide solution was carefully chosen to have optimal rheological properties and exhibit limited shear and oxidative degradation. The sequence of transitions to turbulence found here is notably different from that observed previously for either Newtonian or low-elasticity fluids. As El approaches order 1, laminar and turbulent flows are separated by only two coherent flow states: Standing vortices and disordered rotating standing waves. In contrast to our experiments at lower El, we also observe flow state hysteresis. In addition, the final turbulent flow state was not turbulent Taylor vortices (TTV) as seen with Newtonian and weakly elastic fluids, but rather a state we refer to as elastically dominated turbulence, which occurs at a significantly lower Reynolds number than TTV. Stability mappings involving rotation of the outer cylinder show that the flow states and transitions depend on the amount of counter- or co-rotation. As the degree of counter rotation increased, greater deviations from Newtonian and low El behavior were found, due to the presence of a nodal surface that changes the characteristic length scale of the flow.