The Collaborative International Dictionary of English v.0.48:

Greatness \Great"ness\, n. [AS. gre['a]tnes.]
   1. The state, condition, or quality of being great; as,
      greatness of size, greatness of mind, power, etc.
      [1913 Webster]

   2. Pride; haughtiness. [Obs.]
      [1913 Webster]

            It is not of pride or greatness that he cometh not
            aboard your ships.                    --Bacon.
      [1913 Webster]

WordNet (r) 3.0 (2006):

greatness
    n 1: the property possessed by something or someone of
         outstanding importance or eminence [syn: greatness,
         illustriousness]
    2: unusual largeness in size or extent or number [syn:
       enormousness, grandness, greatness, immenseness,
       immensity, sizeableness, vastness, wideness]