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.
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