Although the viscosity of a fluid ranges over several orders of magnitude and is extremely sensitive to microscopic structure and molecular interactions, it has been conjectured that its (opportunely normalized) minimum displays a universal value which is experimentally approached in strongly coupled fluids such as the quark-gluon plasma. At the same time, recent findings suggest that hydrodynamics could serve as a universal attractor even when the deformation gradients are large and that dissipative transport coefficients, such as viscosity, could still display a universal behavior far from equilibrium. Motivated by these observations, we consider the real-time dissipative dynamics of several holographic models under large shear deformations. In all the cases considered, we observe that at late time both the viscosity-entropy density ratio and the dimensionless ratio between energy density and entropy density approach a constant value. Whenever the shear rate in units of the energy density is small at late time, these values coincide with the expectations from near equilibrium hydrodynamics. Surprisingly, even when this is not the case, and the system at late time is far from equilibrium, the viscosity-to-entropy ratio approaches a constant which decreases monotonically with the dimensionless shear rate and can be parametrically smaller than the hydrodynamic result.