We provide the first characterization of the nonlinear and time dependent rheologic response of viscoelastic bottom-up holographic models. More precisely, we perform oscillatory shear tests in holographic massive gravity theories with finite elastic response, focusing on the large amplitude oscillatory shear (LAOS) regime. The characterization of these systems is done using several techniques: (i) the Lissajous figures, (ii) the Fourier analysis of the stress signal, (iii) the Pipkin diagram and (iv) the dependence of the storage and loss moduli on the amplitude of the applied strain. We find substantial evidence for a strong strain stiffening mechanism, typical of hyperelastic materials such as rubbers and complex polymers. This indicates that the holographic models considered are not a good description for rigid metals, where strain stiffening is not commonly observed. Additionally, a crossover between a viscoelastic liquid regime at small graviton mass (compared to the temperature scale), and a viscoelastic solid regime at large values is observed. Finally, we discuss the relevance of our results for soft matter and for the understanding of the widely used homogeneous holographic models with broken translations.