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]

Moby Thesaurus II by Grady Ward, 1.0:

151 Moby Thesaurus words for "greatness":
   accomplishment, ampleness, amplitude, area, ascendancy, authority,
   bigheartedness, bigness, body, breadth, broadness, brooding, bulk,
   caliber, capaciousness, chivalrousness, chivalry,
   comprehensiveness, consequence, conspicuousness, copiousness,
   coverage, covering, deanship, depth, diameter, dignity, dimension,
   dimensions, distinction, elevation, eminence, errantry, esteem,
   exaltation, excellence, expanse, expansion, expansiveness,
   extension, extensiveness, extent, fame, favor, gauge, generosity,
   generousness, gestation, girth, glory, grandeur, grandiosity,
   grandness, gravidity, gravidness, great heart, greatheartedness,
   greatness of heart, heaviness, height, heroism, high mightiness,
   high-mindedness, honor, idealism, importance, incomparability,
   incubation, influence, influentialness, inimitability,
   knight-errantry, knightliness, largeheartedness, largeness, lead,
   length, liberality, liberalness, loftiness, magnanimity,
   magnanimousness, magnitude, majority, mark, mass, measure,
   measurement, memorability, moment, nobility, noble-mindedness,
   nobleness, notability, note, noteworthiness, one-upmanship,
   openhandedness, outstandingness, precedence, predominance,
   predomination, preeminence, pregnancy, preponderance, prepotence,
   prepotency, prerogative, pressure, prestige, princeliness,
   priority, privilege, profundity, prominence, proportion,
   proportions, radius, range, rank, reach, remarkableness, renown,
   reputation, repute, right-of-way, salience, scale, scope,
   seniority, significance, sitting, size, skill, spaciousness,
   spread, stature, sublimity, success, supereminence, superiority,
   tallness, the family way, transcendence, transcendency, virtuosity,
   volume, weight, weightiness, wideness, width