Note that you need a load impedance, to keep the waveform always pushed down towards zero -- otherwise the voltage rides over the valleys without dropping, and you don't get nearly as much AC output.
If we put a load resistor on there, we lose as much signal power; same as a resistor-biased class A amplifier.
A resistor isn't strictly required, but this is representative of how the method yields fairly poor efficiency.
If we cascade a bunch of stages, pretty quickly the signal power goes away and we wonder what we are even trying to do...
If we can afford amplification, it's still quite reasonable though. And indeed, if we use nonlinear amplifying devices, we can use two in parallel, bias them somewhere into cutoff, and drive them so that they conduct on opposite phases of the input -- the two, FET gates say, are driven oppositely, but their drains are connected in parallel, giving an active full wave rectification, as it were.
What about waveform quality? It works nicely for one stage, but you can easily see that the second stage will be "flipping" the spiky valley into another peak, alongside a smooth sinus peak. It's not at all repetitive. And the 3rd stage will be even weirder, etc. This method doesn't work very well in general, but if we can afford to filter out the higher harmonics, we can smooth that negative valley into another sinus hump, and when paired with the positive hump, it looks alright again.
How much filtering can we afford? If the input signal has a frequency range of less than half an octave, then the doubled output will have a range of less than one octave, and we can cleanly select the doubled fundamental while rejecting all higher harmonics.
We can generalize this further to any harmonic distortion network and filter; say we have an amplifier with a very narrow conduction angle, i.e. it makes a low duty cycle, spiky waveform from a sine wave input -- if we drive a 3rd or even 5th or higher harmonic bandpass filter with this clipped signal, we can make the filter resonate. The filtering won't be perfect, so there will be some amplitude modulation -- the signal rings down between hits, it's like striking a bell at a subharmonic rate. We can make the filter arbitrarily tight to fix this, or we can pair it with an amplitude limiter.
These frequency multiplier circuits were in common use back in the days of vacuum tubes -- when a radio's crystal might be only a few MHz yet a stable VHF or higher carrier is needed, or to convert signals between bands for various purposes, or to widen a signal's bandwidth (FM modulation is difficult; a phase modulator is simple but gives very little deviation; by multiplying it up, then shifting it back down with hetrodyning, wideband (broadcast) FM can be made).
Tim
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