The Collaborative International Dictionary of English v.0.48:

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.
      [1913 Webster]

WordNet (r) 3.0 (2006):

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]

Moby Thesaurus II by Grady Ward, 1.0:

21 Moby Thesaurus words for "deeply":
   acutely, completely, deep, emotionally, entirely, extremely,
   gravely, greatly, heavily, intensely, irrevocably, keenly,
   passionately, powerfully, profoundly, seriously, severely,
   strongly, thoroughly, unreservedly, very much