How often do you have to move the outbreak marker in a chain reaction?
You move it once per outbreak, chain reaction or otherwise.

The rules don't seem ambiguous to me:


  When a chain reaction outbreak occurs, first move the outbreaks marker
  forward 1 space. Then, place cubes as above, except do not add a cube to
  cities that have already had an outbreak (or a chain reaction outbreak) as
  part of resolving the current Infection card.


You have to do all of this - moving the outbreak marker, and placing cubes - for each city that has a chain reaction outbreak.

Also, it says "when a chain reaction outbreak occurs" - as in, one outbreak. If it meant you only moved the marker one step even when there are many chain reaction outbreaks, it would presumably say "one or more" or otherwise clarify.