Why don't infinitely many sinusoidals exist in an oscillator circuit?
A required condition for oscillation in a linear oscillator is that the round trip gain is > 1 (and it will become equal to 1 when oscillation begins and nonlinearities kick in).

Generally we make a narrowband oscillator by making the feedback circuit selective --- it should have low loss for a very narrow band of frequencies, but high loss for other frequencies. 

However, no oscillator is perfectly single frequency. There is always some phase noise that results in an output spectrum with some spread around the center frequency. Since frequency is a continuous value, in some sense there are an "infinite number" of frequencies present --- they're just tightly bunched around the center frequency of the oscillator.