How much does an impact reduce helmet's ability to protect? I read the answers to "When or how often should I replace my bike helmet?", but unfortunately none of the answers there cites any references.

Have there been any studies quantifying the claim of reduced protection after single impact, e.g. in terms of increased risk of injury? 

It's easy to see reasons why manufacturers would want you to believe any impact kills the helmet - it's hard to set limits on potential damage, and buying a new helmet means additional profit. 

Horse riding helmets come with similar warnings, but my first one seemed to survive several years while I was learning show jumping - including way more falls than I could have afforded new helmets.
There is a lot of good information about this here. One paragraph from the article outlines the basic idea of this:


  Crushing the cell walls destroys the impact management ability for most stiff foams, so the helmet has to be replaced after a single impact. The crushing is not always visible and can be hidden by the outer shell. The foam can also recover some of its thickness over a period of hours, but not its ability to manage impact. Crushed and partially recovered one-use foam will feel rubbery and soft. Experts measure the foam thickness carefully for crush, but for consumers the recommendation has to be "replace after every impact." Often the damage is not apparent in a casual inspection, even when there are cracks in the foam. And the rider often underestimates the impact because crushing of the foam dampened the shock. 


It goes on to explain the different types of technologies (with some links to scientific information) used in bike helmets. While your helmet survived all the falls, it was most likely offering you less and less protection from the falls.