  The Smallest Scale of Hierarchy (SSH) survey is an ongoing strategic large
program at the Large Binocular Telescope, aimed at the detection of faint
stellar streams and satellites around 45 late-type dwarf galaxies located in
the Local Universe within $\simeq$10 Mpc. SSH exploits the wide-field, deep
photometry provided by the Large Binocular Cameras in the two wide filters $g$
and $r$. This paper describes the survey, its goals, and the observational and
data reduction strategies. We present preliminary scientific results for five
representative cases (UGC 12613, NGC 2366, UGC 685, NGC 5477 and UGC 4426)
covering the whole distance range spanned by the SSH targets. We reach a
surface brightness limit as faint as $\mu(r)\sim$ 31 mag arcsec$^{-2}$ both for
targets closer than 4$-$5 Mpc, which are resolved into individual stars, and
for more distant targets through the diffuse light. Our analysis reveals the
presence of extended low surface brightness stellar envelopes around the
dwarfs, reaching farther out than what traced by the integrated light, and as
far out as, or even beyond, the observed HI disk. Stellar streams, arcs, and
peculiar features are detected in some cases, indicating possible perturbation,
accretion, or merging events. We also report on the discovery of an extreme
case of Ultra Diffuse Galaxy ($\mu_g(0)=27.9$~mag/arcsec$^2$) in the background
of one of our targets, to illustrate the power of the survey in revealing
extremely low surface brightness systems.
