  Here we report on the formation of two and three magnon bound states in the
quasi-one-dimensional antiferromagnet $\alpha$-NaMnO$_2$, where the single-ion,
uniaxial anisotropy inherent to the Mn$^{3+}$ ions in this material provides a
binding mechanism capable of stabilizing higher order magnon bound states.
While such states have long remained elusive in studies of antiferromagnetic
chains, neutron scattering data presented here demonstrate that higher order
$n>2$ composite magnons exist, and, specifically, that a weak three-magnon
bound state is detected below the antiferromagnetic ordering transition of
NaMnO$_2$. We corroborate our findings with exact numerical simulations of a
one-dimensional Heisenberg chain with easy-axis anisotropy using matrix-product
state techniques, finding a good quantitative agreement with the experiment.
These results establish $\alpha$-NaMnO$_2$ as a unique platform for exploring
the dynamics of composite magnon states inherent to a classical
antiferromagnetic spin chain with Ising-like single ion anisotropy.
