How many times is it safe to reheat chicken?

Taken out of context, that doesn't make much sense, but I can think of several similar claims you might have heard - all of which are true and most can be found on the USDA fact sheet and the sites of most other agencies:

• Perishable food should not be left in the "danger zone" (4.4° C - 60° C or 40° F to 140° F) (room temperature) for more than a total of 2 hours; 1 hour if the temperature is above 32° C / 90° F (mouth/gut temperature).

• Freezing does not reset this countdown, it only stops it temporarily. Thus, it is not a safe practice to thaw and refreeze food several times (especially if it's being brought above refrigerator temperatures).

• Cooking does reset the countdown, but unless you are in a completely sterile environment (hint: you aren't), you still have to keep food out of the danger zone after cooking, which is why guidelines say to refrigerate within 2 hours.

• Most people reheat their food to "warm" or even "hot" but not cooking temperature. Unless you reheat to an internal temperature of 74° C / 165° F, then all that time in the danger zone after cooking is cumulative. So if you do what most people do and reheat food to 40°-50° C, a nice comfortable eating temperature, then you'd better either (a) only reheat the portion you plan to eat, and/or (b) throw out any uneaten portion instead of putting it back in the fridge.

Really it's all about the cumulative amount of time, since being fully cooked, that the food has spent in the danger zone. Theoretically, if you boiled the same piece of chicken every 12 hours and quick-froze it immediately afterward, it would stay safe indefinitely. It would also have no flavour or texture left, so I don't recommend it.

Since most people don't carefully measure the internal temperature of food when they reheat it, we have the "best practice" of only reheating perishable food once after it's been cooked, because otherwise it gets really hard to estimate how much time is left on that metaphorical clock. And, obviously, don't leave it in the fridge for more than a few days, because it can still spoil in there, just slower.