  Discrete breathers with purely anharmonic short-range interaction potentials
localize super-exponentially becoming compact-like. We analyze their spatial
localization properties and their dynamical stability. Several branches of
solutions are identified. One of them connects to the well-known Page and
Sievers-Takeno lattice modes, another one connects with the compacton solutions
of Rosenau. The absence of linear dispersion allows for extremely long-lived
time-quasiperiodic localized excitations. Adding long-range anharmonic
interactions leads to an extreme case of competition between length scales
defining the spatial breather localization. We show that short- and long-range
interaction terms competition results in the appearance of several
characteristic cross-over lengths and essentially breaks the concept of
compactness of the corresponding discrete breathers.
