A couple of days after I posted this to stackexchange, no one's answered:

I take the problem of cumulants to be this: given a sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$ of real numbers, is it the sequence of cumulants of some probability distribution? In one sense, this is trivially equivalent to the problem of moments: the $n$th moment is a polynomial in the first $n$ cumulants and vice-versa. But cumulant sequences have a nice property that moment sequences don't have: the set of all such sequences is closed under addition. So draw a ray out from the origin $(0,0,0,\ldots)$. If the ray bumps into a cumulant sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$, then $2(\kappa_1,\kappa_2,\kappa_3,\ldots),3(\kappa_1,\kappa_2,\kappa_3,\ldots),\ldots$ are also cumulant sequences.

For infinitely divisible distributions, for every real $t\ge 0$, the sequence $t(\kappa_1,\kappa_2,\kappa_3,\ldots)$ is a cumulant sequence.

Besides the nonnegative integers and the nonnegative reals, there are other sets of nonnegative reals closed under addition.

For which sets $T$ of nonnegative reals that are closed under addition is it the case that for some cumulant sequence $(\kappa_1,\kappa_2,\kappa_3,\ldots)$, for every real $t\ge 0$, $$ t\in T \quad\text{iff}\quad t(\kappa_1,\kappa_2,\kappa_3,\ldots)\text{ is a cumulant sequence ?} $$

I'm guessing only closed sets, but there should be more than that to say about it, I would think. Even describing or classifying the (topologically) closed subsets of $\mathbb{R}^+$ that are closed under addition might be something of a chore, and I suspect many of them are not of this form.

That moderately complicated things might happen is at least hinted at by the case of $p\times p$ Wishart matrices, for which the number of degrees of freedom can be anywhere in the set $\lbrace 0,\dots,p-1\rbrace\cup (p-1,\infty)$. But that's for matrix-valued, rather than real-valued, random variables, so it's at most a hint.