An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by A. Grams in the seventies with a neat counterexample. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams’ first construction. In the first part of this paper, we generalize Grams’ construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct a new class of atomic semigroup rings without the ACCP.