A portion of the thesis of my last Ph.D. student, Lynda Danielson, involved a consideration of the following question: when are all iterates of an irreducible polynomial also irreducible? (By the iterates of a polynomial one means the repeated composites of the polynomial with itself; irreducible means that the polynomial can not be factored.) In the thesis, a surprising connection was found between this question and the Beal Conjecture in number theory. Several years later, an anonymous referee showed that there was also a connection with the ABC Conjecture. In this expository talk, I will briefly discuss these two important conjectures and explain how the connections with iterated polynomials arise.