  We consider damped and forced discrete nonlinear Schr\"odinger equations on
the lattice $\mathbb{Z}$. First we establish the existence of periodic and
quasiperiodic breather solutions for periodic and quasiperiodic driving,
respectively. Notably, quasiperiodic breathers cannot exist in the system
without damping and driving. Afterwards the existence of a global uniform
attractor for the dissipative dynamics of the system is shown. For strong
dissipation we prove that the global uniform attractor has finite fractal
dimension and consists of a single trajectory that is confined to a finite
dimensional subspace of the infinite dimensional phase space, attracting any
bounded set in phase space exponentially fast. Conclusively, for strong damping
and periodic (quasiperiodic) forcing the single periodic (quasiperiodic)
breather solution possesses a finite number of modes and is exponentially
stable.
