Oscillatory rheometry has been widely used in bulk rheological measurements of complex fluids such as polymer solutions and melts. Despite recent progress on bulk oscillatory rheology, however, the vast majority of single polymer studies has focused on chain dynamics in simple on/off step strain-rate experiments. In order to fully understand dynamic polymer microstructure and to establish connections with bulk rheology, there is a clear need to study the dynamics of single polymers in more realistic, nonidealized model flows with transient forcing functions. In this work, we study the dynamics of single polymers in large amplitude oscillatory extensional (LAOE) flow using experiments and Brownian dynamics (BD) simulations, and we characterize transient polymer stretch, orientation angle, and average unsteady stretch as functions of the flow strength (Weissenberg number, Wi) and probing frequency (Deborah number, De). Small and large amplitude sinusoidal oscillatory extensional flow are generated in a cross-slot microfluidic geometry, which is facilitated by using an automated flow device called the Stokes trap. This approach allows the conformational dynamics of single DNA molecules to be observed in oscillatory extensional flow for long times. In this way, we observe a characteristic periodic motion of polymers in LAOE including compression, rotation, and stretching between the time dependent principal axes of extension and compression. Interestingly, distinct polymer conformations are observed in LAOE that appear to be analogous to buckling instabilities for rigid or semiflexible filaments under compression. Average unsteady polymer extension is further characterized for single polymers in oscillatory extension across a wide range of Wi and De. In the limit of low Wi, average polymer stretch is interpreted using analytical results based on a Hookean dumbbell model, which can be used to define a critical Wi at the linear to nonlinear transition in oscillatory extension. These results reveal the existence of a master curve for average polymer stretch when plotted as a function of an effective Weissenberg number Wi(eff). Experimental results are compared to BD simulations, and we observe good agreement between simulations and experiments for transient and average unsteady dynamics. Finally, average transient dynamics in oscillatory extensional flow are further interpreted in the context of Pipkin space, defined by the two-dimensional space described by Wi and De.