Martin Gardner, The Annotated Alice:
Past tense of the verb to outgribe (it is connected with the old verb to grike or shrike, from which are derived ‘shriek’ and ‘creak’) ‘squeaked’.
Oxford English Dictionary, 2nd ed.:
A factitious word introduced by ‘Lewis Carroll’ (see quot. 18552). (In quot. 1903 used for ‘outdo’, after the style of out-Herod, etc.). Quot. 18551 also occurs in the first verse of ‘Jabberwocky’ in Through the Looking-Glass (1871) i. 21. 1855 ‘L. CARROLL’ Rectory Umbrella & Mischmasch (1932) 139 All mimsy were the borogoves; And the mome raths outgrabe. Ibid. 140 Outgrabe, past tense of the verb to outgribe. (It is connected with the old verb to grike or shrike, from which are derived ‘shriek’ and ‘creak’.) ‘Squeaked.’ 1876 — Hunting of Snark v. 50 The Beaver had counted with scrupulous care, Attending to every word: But it fairly lost heart, and outgrabe in despair, When the third repetition occurred. 1903 Sat. Rev. 7 Feb. 164/1 Deadmanship! wrote..Dr. Shrapnel..; and the word is fit to stir the jealous admiration of Carlyle or even Lewis Carroll. Indeed Dr. Shrapnel ‘outgrabed’ them both.