We investigate the formation of trimers in an infinite one-dimensional lattice model of hard-core particles with single-particle hopping $t$ and and nearest-neighbour two-body $U$ and three-body $V$ interactions of relevance to Rydberg atoms and polar molecules. For sufficiently attractive $Uleq-2t$ and positive $V>0$ a large trimer is stabilized, which persists as $Vrightarrow infty$, while both attractive $Uleq0$ and $Vleq0$ bind a small trimer. The excited state above this small trimer is also bound and has a large extent; its behavior as $Vrightarrow -infty$ resembles that of the large ground-state trimer. These large bound states appear to admit a continuum description. Furthermore, we find that in the limit $V>>t$, $U<-2t$ the bound-state behavior qualitatively evolves with larger $|U|$ from a state described by the scattering of three far separated particles to a state of a compact dimer scattering with a single particle.
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