Everything I've found indicates it's safe to reheat boiled eggs - but nothing I can find indicates how many times. (I assume one should chill them back down to fridge temperature quickly after the first reheating to even have the chance of using them again.)
I don't think there will be a number of times which is safe. Each time you reheat, you increase the risk that you've not killed all the bacteria. Guildlines are usually to only reheat food only once and to make sure you're essentially cooking it through again, i.e. getting it hot enough to kill off any bacteria. (No source specifically but I have a level 2 Food Safety and Hygiene Qualification)
For boiled eggs specifically there is an article here. I'll highlight that eggs are one of the "danger foods" which you should be more careful about.