Let $A$ be a domain. Recall that $A$ is Cohen-Kaplansky (or CK) if
(CK) any nonzero nonunit of $A$ is a product of irreducible elements, and there are only finitely irreducible elements up to multiplication by units.
Consider the following condition on $A$:
(RFFF) the field of fractions $K$ of $A$ is ring-finite over $A$, that is, $K$ is a finitely generated $A$-algebra.
[RFFF stands for "ring-finite field of fractions".]
Clearly, (CK) implies (RFFF).
Naive Conjecture: (RFFF) implies (CK).
Question: Is the Naive Conjecture true?
One easily sees that (RFFF) is equivalent to
(RFFF') there is a nonzero element $a$ in $A$ such that, for any $b$ in $A$, there is a positive integer $n(b)$ such that $b$ divides $a^{n(b)}$.
In particular, the Naive Conjecture holds for unique factorization domains, and also for Dedekind domains.
EDIT A. Thank you very much to Hagen and Bill Dubuque for their answers! Here is an update. From now on, following Kaplansky, I'll call G-domain what I called RFFF-domain above.
$(1)$ Here is how I understand a part of Hagen's answer: Bourbaki, Algèbre Commutative, VI.$6.3$, provides the following example of a G-domain which is not a CK-domain. There exist a field $K$ and a surjective multiplicative monoid morphism $x\mapsto|x|$ from $K$ onto $\mathbb Q_{\ge0}$ satisfying $ |x|=0\iff x=0,\quad|x+y|\le\max(|x|,|y|), $ such that the closed ball of radius one is a local domain $A$ with maximal ideal $\mathfrak m$ equal to the open ball of radius one, the ideals of $A$ being the closed balls of radius $r$ with $0\le r\le1$ and the open balls of radius $r$ with $0 < r\le1$, and the group $A^\times$ of units of $A$ being the sphere of radius one. In particular $0$ and $\mathfrak m$ are the only prime ideals. Thus any nonzero $a$ in $\mathfrak m$ satisfies $A[a^{-1}]=K$, so $A$ is a G-domain. The group $K^\times/A^\times$, being isomorphic to $\mathbb Q_{ > 0}$, is not finitely generated, that is $A$ is not a CK-domain.
$(2)$ In the article
Rings with a finite number of primes, I. S. Cohen and Irving Kaplansky, Trans. Amer. Math. Soc. $60$ ($1946$), $468$-$477$,
it is proved that CK-domains are noetherian (Theorem $6$ p. $471$).
$(3)$ Recall from Bill Dubuque's answer that a domain $A$ is a noetherian G-domains if and only if $A$ has only a finite number of non-zero prime ideals, all of which are maximal.
$(4)$ The main question which remains open (at least for me) is this:
Is there a noetherian G-domain which is not a CK-domain?
That is:
Is there a noetherian domain $A$ such that
$\bullet\ A$ has only a finite number of non-zero prime ideals, all of which are maximal,
$\bullet\ A$ has infinitely many (association classes of) irreducibles?