Can you give examples of oscillatory motions which aren't harmonic?

Anharmonicity can be found in (for example) an atom, which consists of a positively charged nucleus surrounded by a negatively charged electronic cloud, it experiences a displacement between the center of mass of the nucleus and the electronic cloud when an electric field is present. The amount of that displacement, called the electric dipole moment, is related linearly to the applied field for small fields, but as the magnitude of the field is increased, the field-dipole moment relationship becomes nonlinear, just as in the mechanical system.

Further examples of anharmonic oscillators include the large-angle pendulum, which exhibits chaotic behavior as a result of its anharmonicity; nonequilibrium semiconductors that possess a large hot carrier population, which exhibit nonlinear behaviors of various types related to the effective mass of the carriers; and ionospheric plasmas, which also exhibit nonlinear behavior based on the anharmonicity of the plasma. In fact, virtually all oscillators become anharmonic when their pump amplitude increases beyond some threshold, and as a result it is necessary to use nonlinear equations of motion to describe their behavior.

Anharmonicity also plays a role in lattice and molecular vibrations, in quantum oscillations.