Let $R$ be a commutative ring. Let's say that the Jacobson radical $J(R)$ of $R$ is *uninteresting* if

$J(R)$ coincides with the nilradical, or

$J(R)$ is the intersection of a finite number of maximal ideals.

It seems as if most rings used in algebraic geometry have uninteresting Jacbson radical:

Every finitely-generated commutative algebra over a field or over a Dedekind domain is Jacobson, so its Jacobson radical coincides with its nilradical, and so is uninteresting by (1).

Every local or semilocal commutative ring has finitely many maximal ideals, and so has uninteresting Jacobson radical by (2).

For a nonuninteresting example, take the localization $R = \mathbb Z[x]_S$ where $S = \{f(x) \in \mathbb Z[x] \mid f(0) = 1\}$. The maximal ideals of $R$ are of the form $(p,x)$ where $p \in \operatorname{Spec} \mathbb Z$. So the Jacobson radical is $(x)$, which is not uninteresting.

But this example seems rather artificial to me; for example I don't know anywhere a ring like this would show up in algebraic geometry.