Deep-six destroy or dispose of (something) irretrievably, perhaps from the custom of burial at sea at a depth of six fathoms.
Deep South the south-eastern region of the US regarded as embodying traditional Southern culture and traditions.
Deep Throat codename (from the title of a pornographic film, 1972) given in the Watergate affair to the journalists' principal anonymous informant; from this, deep throat is used to mean a person working for an organization who anonymously supplies information on misconduct to an outside source. The original Deep Throat has never been publicly identified.
See also beauty is only skin deep, still waters run deep.