The Collaborative International Dictionary of English v.0.48:

Infinitely \In"fi*nite*ly\, adv.
   1. Without bounds or limits; beyond or below assignable
      limits; as, an infinitely large or infinitely small
      quantity.
      [1913 Webster]

   2. Very; exceedingly; vastly; highly; extremely. "Infinitely
      pleased." --Dryden.
      [1913 Webster]

WordNet (r) 3.0 (2006):

infinitely
    adv 1: without bounds; "he is infinitely wealthy" [syn:
           boundlessly, immeasurably, infinitely]
    2: continuing forever without end; "there are infinitely many
       possibilities" [syn: infinitely, endlessly] [ant:
       finitely]

Moby Thesaurus II by Grady Ward, 1.0:

83 Moby Thesaurus words for "infinitely":
   a outrance, abroad, absolutely, abundantly, acervatim,
   ad infinitum, all out, beyond all bounds, beyond compare,
   beyond comparison, beyond measure, boundlessly, broadly,
   ceaselessly, completely, constantly, continually, copiously,
   countlessly, dead, downright, endlessly, essentially, eternally,
   everlastingly, extensively, extremely, far and near, far and wide,
   flat out, forever, fundamentally, illimitably, immeasurably,
   immensely, in crowds, in heaps, in perpetuity, in the extreme,
   in throngs, incalculably, incessantly, incomprehensibly,
   indefinitely, indestructibly, inexhaustibly, innumerably,
   interminably, limitlessly, measurelessly, most, multitudinously,
   never-endingly, no end, numerously, on all sides, on every side,
   perdurably, perennially, perfectly, permanently, perpetually,
   prodigally, profusely, purely, radically, right and left, steadily,
   thickly, time without end, to infinity, totally, unceasingly,
   unconditionally, unendingly, unequivocally, utterly, vastly,
   widely, with a vengeance, without end, without limit,
   world without end