Ah yes, just run a square wave at twice the range of human hearing, then use a combination of filters to extract the desired ranges. Unfortunately you'd also need an infinite arrangement of filters.
Here's a synth using the technique:
I won't even make you write the code that generates the stream, I'll just require that one sample.
If you can do it, I'll give you $1000 out of my own pocket.
Given Nyquist, a non-infinite series of [non-trivial power of 2] nodes will do the job just fine.
Audio FFT/iFFT processing typically uses 512 or 1024 bins, although sometimes you can get away with fewer.
In practice, iFFT doesn't use discrete oscillators for resynthesis, because the whole point of iFFTs is to limit the amount of work you have to do.
But there's no reason in theory a limited number of oscillators couldn't do the job. (I've done resynthesis like this in SuperCollider when I wanted special effects that FFTs can't produce.)
I doubt WA would be fast enough on most machines, but it would probably be possible - if rather dumb - on powerful hardware.