The Collaborative International Dictionary of English v.0.48:

Highness \High"ness\, n. [AS. he['a]hnes.]
   1. The state of being high; elevation; loftiness.
      [1913 Webster]

   2. A title of honor given to kings, princes, or other persons
      of rank; as, His Royal Highness. --Shak.
      [1913 Webster]

WordNet (r) 3.0 (2006):

Highness
    n 1: (Your Highness or His Highness or Her Highness) title used
         to address a royal person
    2: the quality of being high or lofty [syn: highness,
       loftiness] [ant: lowness]
    3: a high degree (of amount or force etc.); "responsible for the
       highness of the rates"