So if an outbreak occurs, I do move the outbreak marker one forward. If this triggers a chain reaction in one city, I move it forward once more. But what happens if I have several chain reactions? Would I have to move the outbreak marker up to 8 times? The rule book is a bit ambiguous about this.
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.