Yes, it can have an oscillator that even for small amplitudes is not simple harmonic. That is it can have a nonlinear restoring force in an oscillator even at arbitrarily small amplitudes.
The motion of a body is said to be periodic if it passes through similar conditions after equal intervals of time. Oscillations of a particle are said to be harmonic if its displacement, at any instant, can be expressed as a function of sine or cosine of an angle depending upon time.
Harmonic oscillation is a general term applied to oscillations in which a particle may oscillate within unequal limits about the mean position. A special case of such oscillation in which limits of oscillations on either side of the neutral position are equal is known as simple harmonic oscillations. So, simple harmonic motion is the motion in which the restoring force is proportional to displacement from the mean position and opposes its increase. Therefore if a nonlinear restoring force acts on the oscillator, then the oscillation of the system will not harmonic even for small amplitudes.