outgrow v.

in Old English a common prefix with nouns, adjectives, adverbs, and verbs, "out, outward, outer; forth, away," from out (adv.). The use was even more common in Middle English, and also with the senses "outer, outside, on the outside, from without, external, externally; apart; greatly, extremely; completely, thoroughly, to completion." Other senses of out that extended into the use as a prefix include "beyond the surface or limits; to the utmost degree; to an explicit resolution."

In composition out has either its ordinary adverbial sense, as in outcast, outcome, outlook, etc., or a prepositional force, as in outdoors, or forms transitive verbs denoting a going beyond or surpassing of the object of the verb, in doing the act expressed by the word to which it is prefixed, as in outrun, outshine, outvenom, etc. In the last use especially out may be used with almost any noun or verb. [Century Dictionary]

Old English growan (of plants) "to flourish, increase, develop, get bigger" (class VII strong verb; past tense greow, past participle growen), from Proto-Germanic *gro- (source also of Old Norse groa "to grow" (of vegetation), Old Frisian groia, Dutch groeien, Old High German gruoen), from PIE root *ghre- "to grow, become green" (see grass). Applied in Middle English to human beings (c. 1300) and animals (early 15c.) and their parts, supplanting Old English weaxan (see wax (v.)) in the general sense of "to increase." Transitive sense "cause to grow" is from 1774. To grow on "gain in the estimation of" is from 1712.

Have you ever heard anything about God, Topsy? ... Do you know who made you?" "Nobody, as I knows on," said the child. ... "I spect I grow'd. Don't think nobody never made me." [Harriet B. Stowe, "Uncle Tom's Cabin," 1851]