From The Collaborative International Dictionary of English v.0.48 [gcide]:
Deeply \Deep"ly\, adv.
1. At or to a great depth; far below the surface; as, to sink
deeply.
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2. Profoundly; thoroughly; not superficially; in a high
degree; intensely; as, deeply skilled in ethics.
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He had deeply offended both his nobles and people.
--Bacon.
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He sighed deeply in his spirit. --Mark viii.
12.
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3. Very; with a tendency to darkness of color.
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The deeply red juice of buckthorn berries. --Boyle.
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4. Gravely; with low or deep tone; as, a deeply toned
instrument.
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5. With profound skill; with art or intricacy; as, a deeply
laid plot or intrigue.
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From WordNet (r) 3.0 (2006) [wn]:
deeply
adv 1: to a great depth psychologically; "They felt the loss
deeply" [syn: profoundly, deeply]
2: to a great depth;far down; "dived deeply"; "dug deep" [syn:
deeply, deep]