Small but powerful speakers
Standard speaker maximum power is limited by coil resistance (more resistance - more heat) and mechanical durability.

High power at small size can be archieved by removing low frequencies from signal and heat  from coil resistance is power limiting factor. 

At high frequency - membrane moves faster than at low frequency, so it needs more acceleration (and more power can be used at speaker same size).

For low frequencies - mechanical durability matters a lot. You also need bigger surface to use that much power for just pushing air (and move membrane mass).


  To move more air does not mean to have a bigger speaker? Something
  else is also in play. What is it?


First thing to consider is amount/volume of air moved by speaker membrane (Vd speaker parameter) This is related with membrane surface area (Sd speaker parameter) and maximum membrane travel (Xmax speaker parameter).

Small speaker needs higher membrane travel to push same volume of air. It's easier to build a speaker with low travel and high surface, than small surface and long travel. Thats why low cost speakers are bigger than more expensive speakers.

Second thing is frequency range of speaker and resonance effects. This is more complicated.


  So for my analog 20W stereo amp, what is the smallest possible full
  range speaker I can get?


There is no answer for that question. It depends on what you mean by "full range" and what you mean by "size" (is it speaker diameter or box size). You can look what is available on the market (computer speakers with similar power, mini stereo systems etc). That should give you idea what is possible and what is not.


  And how do companies, like Libratone and Bose, create small speakers
  with high power(70W, 100W), though they do separate it with
  crossovers?


Small, very high power speakers have narrow frequency range. You can buy 100W 3000-20000Hz speaker with 100W power and 1 inch membrane or speaker that has 300-3000Hz frequency range, 80W and 4 inch diameter but this is not full range.