  We investigate the problem of $N$ identical bosons that are coupled to an
impurity particle with infinite mass. For non-interacting bosons, we show that
a dynamical impurity-boson interaction, mediated by a closed-channel dimer, can
induce an effective boson-boson repulsion which strongly modifies the bound
states consisting of the impurity and $N$ bosons. In particular, we demonstrate
the existence of two universal 'multi-body' resonances, where all multi-body
bound states involving any $N$ emerge and disappear. The first multi-body
resonance corresponds to infinite impurity-boson scattering length, $a\to
+\infty$, while the second corresponds to the critical scattering length
$a^*>0$ beyond which the trimer ($N=2$ bound state) ceases to exist. Crucially,
we show that the existence of $a^*$ ensures that the ground-state energy in the
multi-body bound-state region, $\infty>a> a^*$, is bounded from below, with a
bound that is independent of $N$. Thus, even though the impurity can support
multi-body bound states, they become increasingly fragile beyond the dimer
state. This has implications for the nature of the Bose polaron currently being
studied in cold-atom experiments.
