"to ask alms," especially to do so habitually as one's way of life, c. 1200, of uncertain origin. Perhaps from the rare Old English bedecian "to beg," from Proto-Germanic *beth-. Or from Anglo-French begger, a back-formation from Old French noun begart (see beggar (n.)) and ultimately from Beguine, which OED considers "perhaps the most likely derivation." The Old English word for "beg" was wædlian, from wædl "poverty." Related: Begged; begging.
Meaning "ask for" (a favor, etc.) is by 1520s. As a courteous mode of asking (beg pardon, etc.), attested by c. 1600. Of dogs, 1762. To beg the question (1580s) translates Latin petitio principii, and means "to assume something that hasn't been proven as a basis of one's argument," thus "asking" one's opponent to give something unearned, though more of the nature of taking it for granted without warrant. To beg off (something) "obtain release from by entreaty" is from 1741.