Non-equilibrium molecular-dynamics simulations performed on the Lennard-Jones, LJ, liquid near its triple point reveal first and second Newtonian regimes, extreme shear thinning and shear thickening. Extensive computations at low shear rates reveal a departure from the linear relationship between the viscosity and the square root of the shear rate, that is observed under moderately shear thinning conditions. Similar deviations for the energy and pressure are observed with respect to ^1.5. Long-range order develops in the liquid during extreme shear thinning, characterised by effectively straight lines of molecules packing along the shear flow lines in a triangular lattice. The number density, the diagonal pressure tensor components, the shear rigidity moduli, the self-diffusion coefficients, the time correlation functions all become highly anisotropic in this regime, reflecting the largely athermal motion of the strings governed by the magnitude of the shear rate. Local parity breaking structural asymmetry with lifetimes comparable to the simulation are observed at the highest shear rates considered, leading eventually to structural disorder and associated shear thickening via dislocations in the triangular lattice at a shear rate of ca. 100 LJ reduced units.