In micro/nanometer scale mechanical resonators, constructive utilization of intentional nonlinearity has suggested ways to leverage beneficial nonlinear characteristics in their design for various applications. Previous studies have also shown that the geometric nonlinearity is effectively implemented and tailored through integration of nonlinear couplings to an otherwise linear microcantilever. Here, we demonstrate experimentally a nonlinear micromechanical resonator consisting of a silicon microcantilever axially constrained by a polymer attachment exhibiting a strong nonlinear hardening behavior not only in its first flexural mode but also in higher modes. A theoretical model representing the system with geometrically nonlinear stiffness and damping is analyzed by the method of multiple scales, which is favorably validated by good agreement with experimentally obtained nonlinear responses.