Did you mean fewest or most? Fewest- don’t drop them at all. Most- go into space. At zero gravity, if you are careful not to put enough force on them to set them moving into something at enough speed to break them, you could drop them a lot of times.
Don't know if this is meant to be a trick or joke question - but if you hardboil eggs and drop them from a very tiny height without added impact other than gravity - you could theoretically drop them several times without their breaking. No idea why anyone would want to do that though