Barkhausen Criteria. Is unity loop gain required?
If the loop gain of an oscillator would be greater than one, then unless the circuit is in perfect balance such that the amplitude is precisely zero, oscillations will grow until either they have gotten so big as to swallow up the entire universe (unlikely), or until the circuits in the loop are unable to maintain a gain greater than one for an entire cycle (much more likely).  If the gain of the system drops with average amplitude but remains constant throughout each cycle, the oscillator will produce a clean output.  If, as is somewhat more typical, the gain drops as a function of instantaneous amplitude, the variation in gain throughout each cycle will cause distortion.  Generally, the closer the "maximum" gain is to unity, the less variation there will be in gain during each cycle and the cleaner the output will be.

Note that a relaxation oscillator represents an extreme case of variable loop gain; in an ideal relaxation oscillator, the loop gain will be zero for much of the cycle, and infinite during an infinitesimal portion of the cycle.  The "unity gain" requirement applies for relaxation oscillators just as for other oscillators, but the fact that the gain is very high for a very small portion of the cycle makes them numerically far less useful than for resonant oscillators.