What is an example of a nonideal oscillator?

A real molecule, modeled by an anharmonic oscillator combined with a rigid rotator (hence no longer rigid), is a nice example.

Imagine you have a binary molecule that can be modeled like two spheres connected by a real spring.

When the spring is in its equilibrium position without disturbing it, let's assume it's stiff enough to stay in place (while it's on a table or something). As you stretch it out, you decide to let it go at some arbitrary moment.

Now imagine there's no table after you let the molecular spring oscillate.

It's not going to just oscillate horizontally, it's also going to wag up and down and rotate because it's not just a harmonic oscillator, it's an anharmonic oscillator AND a nonrigid rotator---a realistic spring in 3D space.

The further away each atom is from the other, the more easily the vertical wagging can happen because the stiffness decreases in a sense; it's the same idea as torque becoming greater as the distance to the axis of rotation increases. The easier the vertical wagging, the more easily the energy is dissipated into the air from the wagging, and the less energy is available for the horizontal oscillation.

Overall, you get a slightly smaller frequency at a greater amplitude, and you get centrifugal distortion of the frequencies as well from ro-vibrational coupling.

Also, as the amplitude increases, the anharmonic oscillator slows down, and eventually, it'll stop oscillating, instead of constantly oscillating forever (mainly because of the spring, not as much because of air resistance).

You can see more of that by looking up the "Damped Harmonic Oscillator".