Any preference which is "irrational" under those axioms, and you already acknowledged preferences of that sort existed.

As an extremely basic example: I could prefer chocolate ice cream over vanilla ice cream, and prefer vanilla ice cream over pistachio ice cream. Under the Von Neumann-Morgenstein axioms, however, I cannot then prefer pistachio to chocolate because that would violate the transitivity axiom. You are correct that there is probably someone out there who holds all three preferences simultaneously. I would call such a person "irrational". Wouldn't you?