  The ground-breaking discovery of the first fully charmed tetraquark state
$X(6900)$ in the $J/\psi J/\psi$ invariant mass distribution by the LHCb
collaboration has inspired intensive theoretical studies. Various
interpretations, such as molecular states, compact tetraquark states, and
coupled-channel effects, have been proposed for these states. Of particular
interest is the ongoing search for the triple-$J/\psi$ state--a fully-charmed
hexaquark state. To deepen our understanding of the triple-$J/\psi$ state and
to guide future experimental searches, we study the triple-$J/\psi$ system in
this work employing the Gaussian expansion method and the $J/\psi J/\psi$
potential parameterized to yield a shallow bound state, as suggested in several
theoretical works. Our results support a triple-$J/\psi$ bound state, even in
cases where the attractive interaction between the two $J/\psi$ mesons is very
weak. Moreover, our analysis implies the Efimov effect in the triple-$J/\psi$
system. In addition, we extend our investigation to the triple-$\Upsilon (1S)$
system and obtain results similar to those for the triple-$J/\psi$ system.
