In at least one telling, no boxes were exchanged on Christmas Day, no incense, no gold, as the Kings didn't arrive until January 6, Epiphany.
Whether we call these three "Kings" or not is open to dispute, as some prefer "Magi," a nickname for "Magicians," although that later sounded too pagan, although "Wizards" was worse (plus they were pagans).
Anyway, in some folks' tellings, there's this fourth King (affectionately known as King Moron) who showed up really belatedly, after the first three had long gone. This tale provides cover for people sending truly late gifts, even after Epiphany is long over.
This fourth King is a bit too much the clown to be admitted into classic tellings, such as Christian hymnals and so forth. Besides, he wouldn't be in the manger scene anyway, so there's nothing to add in this case.