Non-example of UFD

Question

The point of UFD is that element can be factored finitely and uniquely. One non-example is $Z[\sqrt{-5}]$ where there may be non-unique factorization. I wonder if there is any simple example that violates the finiteness requirement, i.e. there is some element that can be factored again and again into non-unit elements.

Answer 1

In the ring of all algebraic integers, every nonzero nonunit is a square, for example. In this case, there aren't any irreducible elements at all.