Well for me, I can't do the same city's quest right after finishing it. I finished the one in Ul'dah and upon trying to initiate the next one it didn't even give me the "You are unable to receive another Archon egg at this time" message. The NPC still had the ! over his head but he just recited the same line as if I still had the quest.

You can however bring eggs back to the hat giving guy and get the same Pristine hat again.