Where do overtones in a 555 generated square wave come from?
I know that a square can be constructed from sine waves.
  However, the 555 does not generate sine waves or multiple frequencies, it generates a single square pulse. So where have these harmonic frequencies come from?


Congratulations on your explanation of what you are seeing and your experimentation.  

The key issue is that not only CAN a square wave be constructed from sine waves, it fundamentally IS a collection of sine waves.
You can generate a square wave by summing appropriate sine waves, but, however you do it, what you arrive at IS a waveform that can be represented by a collection of sine waves.

In ideal circumstances you would not expect to see on the spectrum analyser quite what you show, but impedance matching and a 555 and .... can easily combine to produce a non ideal result.  

A square wave = a summation of \$ f + \frac{3f}{3} + \frac{5f}{5} + \frac{7f}{7} + ...\$ (if my brain has correctly retrieved the relevant long ago stored facts). So you would expect to see every second harmonic, and amplitudes should decrease.