A domain is called atomic if each non-zero and non-invertible elements is a product of irreducible elements.
It can be shown that for an atomic domain $D$ it is equivalent:
That is for atomic domains there is nothing but UFDs.
On the other hand a UFD is always atomic; as a prime is always irreducible.
Thus, if one wants to have something beyond UFDs one needs to leave the realm of domains where each element has a factorization into irreducibles. As Bill Dubuque mentions this is well possible and done. The "largest," that is most inclusive, defintion you seek would be just: "a domain where each irreducible is prime." A name for this is AP-domain (see comment by OP).
In such a domain it is true that every non-zero non-identity element has at most one factorization into irreducibles (up to multiplication by units and reordering).
Note that for a UFD one has the characterization each non-zero non-identity element has exactly one factorization into irreducibles (up to multiplication by units and reordering)
Put differently, in an AP-domain it is true that each element that has a factorization into irreducibles at all has a (essentially) unique factorization.
Such a domain is called an unrestricted UFD by Coykendall and Zafrullah. Thus, every AP-domain is an unrestricted UFD. The converse is not true though as shown by Coykendall and Zafrullah (in contrast to the situation for atomic domains).
A relevant paper is: Jim Coykendall, Muhammad Zafrullah, AP-domains and unique factorization, Journal of Pure and Applied Algebra (2004).