An extrapolation cascadic Newton multigrid (ECNMG) method is proposed for high accuracy numerical solutions of two-dimensional nonlinear Poisson equations, by incorporating the fourth-order compact difference schemes, the extrapolation techniques and the existing Newton multigrid method. A series of grid level dependent computational tolerances are discussed to distribute computational cost on different grids, and an extrapolation interpolation strategy and a bi-quartic polynomial interpolation are used for two fourth-order approximations from current and previous grids to provide an extremely accurate initial guess on the next finer grid, which can greatly reduce the iterations of the Newton multigrid computation for computing an approximation with discretization-level accuracy. Additionally, a completed Richardson extrapolation technique is adopted for the fourth-order computed solution to generate a sixth-order extrapolated solution cheaply. Numerical results of two-dimensional nonlinear Poisson-Boltzmann equations with five different fourth-order compact difference schemes are conducted to demonstrate the new ECNMG algorithm achieve sixth-order accuracy and keep less cost simultaneously, more efficient than the existing Newton-MG method.