We study long-range percolation on the $d$-dimensional hierarchical lattice, in which each possible edge $\{x,y\}$ is included independently at random with inclusion probability $1-\exp ( -β\|x-y\|^{-d-α} )$, where $α>0$ is fixed and $β\geq 0$ is a parameter. This model is known to have a phase transition at some $β_c<\infty$ if and only if $α<d$. We study the model in the regime $α\geq d$, in which $β_c=\infty$, and prove that the susceptibility $χ(β)$ (i.e., the expected volume of the cluster at the origin) satisfies \[ χ(β) = β^{\frac{d}{α- d } - o(1)} \qquad \text{as $β\to \infty$ if $α> d$} \qquad \text{and} \qquad e^{e^{ Θ(β) }} \qquad \text{as $β\to \infty$ if $α= d$.} \] This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that $χ(β)$ grows between exponentially and double-exponentially when $α=d$. Our results imply that analogous results hold for a number of related models including Dyson's hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.