We found an article in the Denver Post (Denver, Colorado, USA) recently:
"Cyclists, rejoice: AAA Colorado has expanded its roadside assistance program to cover bicycles, which means members whose bikes break down during a ride can now call for help.
Under the expanded program, AAA will retrieve members and transport them and their bikes to the destination of their choice for no extra charge , AAA Colorado spokeswoman Wave Dreher said. "*
I think it is a capital wheeze, but whether it is applicable to Europe I don't know. Whether the AA, the RAC, the ADAC, ACE, etc. have the extra capacity to supply more services is debatable, although some of them do charge serious sums of money for their services, for their insurance.
The Dutch motor club, ANWB does offers a bike roadside assistance insurance scheme (translated from Dutch):
Roadside Bicycle Service is roadside assistance for your (electric) bike, road bike or cargobike.
24/7 roadside assistance in the Netherlands.
If the bike cannot be repaired on the spot. The ANWB will bring you to your destination or to a bicycle shop.
The service costs 24 Euros annually. There are discounts. There is more information on the ANWB website: http://www.anwb.nl/wegenwacht/fiets in Dutch only, but it translates well with Google Translator.
If your bike breaks down, there is also a chance that a passing cyclist will offer to help you. It does happen, but not always as our experience in Mainz shows (http://www.crazyguyonabike.com/doc/13861).