As a practical thing, you have to have some sort of AGC or a limiter. Often you have both. An AGC tends to be slow and give you cycle to cycle amplitude variation. A limiter gives you some finite amount of distortion on the output.
You have several options for stabilizing the amplitude:
* The most simple method is to use two anti-parallel diodes across the gain determining resistor - thus performing something like "soft-clipping.
* A bit more efficient (better THD) is a FET which is used as a variable resistor using a control voltage that is derived from the output amplitude (after rectification).
This is a kind of AGC - however, it is NOT true that this AGC loop gives you a "
cycle to cycle amplitude variation". Exactly the
opposite is true because the AGC time constant can/should be selected much larger than one cycle.
* As another option you can accept hard-limiting (power suplly rails) if the frequency-determining network provides enough damping of the third harmonic (lowpass or bandpass filtering). In this case, you can use THIS output node using another opamp which works as a buffer amplifier.
*Further alternatives: Thermistors, light bulb, OTA as resistor or as controllable amplifier.