  We pursue three-body bound states in a one-dimensional tight-binding lattice
described by the Bose-Hubbard model with strong on-site interaction. Apart from
the simple strongly-bound "trimer" state corresponding to all three particles
occupying the same lattice site, we find two novel kinds of weakly-bound
trimers with energies below and above the continuum of scattering states of a
single particle ("monomer") and a bound particle pair ("dimer"). The
corresponding binding mechanism can be inferred from an effective Hamiltonian
in the strong-coupling regime which contains an exchange interaction between
the monomer and dimer. In the limit of very strong on-site interaction, the
exchange-bound trimers attain a universal value of the binding energy. These
phenomena can be observed with cold atoms in optical lattices.
