  The $DDK$ 3-body system is supposed to be bound due to the strongly
attractive interaction between the $D$ meson and the $K$ meson in the isospin
zero channel. The minimum quark content of this 3-body bound state is
$cc\bar{q}\bar{s}$ with $q=u,d$. It will be an explicitly exotic tetraquark
state once discovered. In order to confirm the phenomenological study of the
$DDK$ system, we can refer to lattice QCD as a powerful theoretical tool
parallel to the experiment measurement. In this paper, a 3-body quantization
condition scheme is derived via the non-relativistic effective theory and the
particle-dimer picture in finite volume. Lattice spectrum of this 3-body system
is calculated within the existing model inputs. The spectrum shows various
interesting properties of the $DDK$ system, and it may reveal the nature of the
$D^*(2317)$. This predicated spectrum is expected to be tested in future
lattice simulations.
