Bacterial biofilms in natural and artificial environments perform a wide array of beneficial or detrimental functions and exhibit resistance to physical as well as chemical perturbations. In dynamic environments, where periodic or aperiodic flows over surfaces are involved, biofilms can be subjected to large shear forces. The ability to withstand these forces, which is often attributed to the resilience of the extracellular matrix. This attribute of the extracellular matrix is referred to as viscoelasticity and is a result of self-assembly and cross-linking of multiple polymeric components that are secreted by the microbes. We aim to understand the viscoelastic characteristic of biofilms subjected to large shear forces by performing Large Amplitude Oscillatory Shear (LAOS) experiments on four species of bacterial biofilms: Bacillus subtilis, Comamonas denitrificans, Pseudomonas fluorescens and Pseudomonas aeruginosa. We find that nonlinear viscoelastic measures such as intracycle strain stiffening and intracycle shear thickening for each of the tested species, exhibit subtle or distinct differences in the plot of strain amplitude versus frequency (Pipkin diagram). The biofilms also exhibit variability in the onset of nonlinear behaviour and energy dissipation characteristics, which could be a result of heterogeneity of the extracellular matrix constituents of the different biofilms. The results provide insight into the nonlinear rheological behaviour of biofilms as they are subjected to large strains or strain rates; a situation that is commonly encountered in nature, but rarely investigated.