  In the present work we explore the potential of models of the discrete
nonlinear Schr\"odinger (DNLS) type to support spatially localized and
temporally quasiperiodic solutions on top of a finite background. Such
solutions are rigorously shown to exist in the vicinity of the anti-continuum,
vanishing coupling limit of the model. We then use numerical continuation to
illustrate their persistence for finite coupling, as well as to explore their
spectral stability. We obtain an intricate bifurcation diagram showing a
progression of such solutions from simpler ones bearing single- and two-site
excitations to more complex, multi-site ones with a direct connection of the
branches of the self-focusing and self-defocusing nonlinear regime. We further
probe the variation of the solutions obtained towards the limit of vanishing
frequency for both signs of the nonlinearity. Our analysis is complemented by
exploring the dynamics of the solutions via direct numerical simulations.
