The Collaborative International Dictionary of English v.0.48:

Outdo \Out*do"\ (out*d[=oo]"), v. t. [imp. Outdid
   (out*d[i^]d"); p. p. Outdone (out*d[u^]n"); p. pr. & vb. n.
   Outdoing.]
   To go beyond in performance; to excel; to surpass.
   [1913 Webster]

         An imposture outdoes the original.       --L' Estrange.
   [1913 Webster]

         I grieve to be outdone by Gay.           --Swift.
   [1913 Webster]

   To outdo oneself to surpass one's own previous best
      performance.
      [PJC]

WordNet (r) 3.0 (2006):

outdo
    v 1: be or do something to a greater degree; "her performance
         surpasses that of any other student I know"; "She outdoes
         all other athletes"; "This exceeds all my expectations";
         "This car outperforms all others in its class" [syn:
         surpass, outstrip, outmatch, outgo, exceed,
         outdo, surmount, outperform]
    2: get the better of; "the goal was to best the competition"
       [syn: outdo, outflank, trump, best, scoop]

Moby Thesaurus II by Grady Ward, 1.0:

63 Moby Thesaurus words for "outdo":
   beat, beat all hollow, beat hollow, best, cap, defeat, destroy,
   do in, drub, exceed, excel, fix, hide, hors de combat, lambaste,
   lather, lick, outclass, outfight, outgeneral, outgo, outjump,
   outleap, outmaneuver, outmarch, outpace, outperform, outplay,
   outpoint, outrange, outreach, outride, outrival, outrun, outsail,
   outshine, outstep, outstrip, outvie, outweigh, outwit, overcome,
   overjump, overleap, overmatch, overplay, override, put, ruin,
   settle, skin, skin alive, surpass, thrash, top, transcend, trim,
   triumph over, trounce, trump, undo, whip, worst



The Devil's Dictionary (1881-1906):

OUTDO, v.t.  To make an enemy.