"one more than fifty-nine, twice thirty, six times ten; the number which is one more than fifty-nine; a symbol representing this number;" Middle English sixti, from Old English sixtig, from siex (see six) + -tig (see -ty (1)). Similar formation in Old Norse sextugr, sextögr, sextigir, Old Frisian sextich, Middle Dutch sestig, Dutch zestig, Old High German sehszug, German sechzig.
To do something like sixty "with great force or vigor" is by 1833, American English, but the signification of the sixty is unclear.
"... you know she paints admirably, (all I know is that she daubs like sixty, and calls it painting) ..." [North Carolina Sentinel, Newbern, N.C., July 19, 1833, reprinted "From the Auburn Journal," written in the voice of a common man]
The variant like sixty-six is more recent (by 1853 in North Carolina newspapers) and thus might be a mere embiggening of it.
"one more than sixty-eight; the number which is one more than sixty-eight; a symbol representing this number;" see sixty + nine. In the sexual sense, 1888, as a translation of French faire soixante neuf, literally "to do 69." So called from the resemblance of the persons fitted together to the arrangement of the two numerals.
The phrase sixty-four dollar question "crucial question, most difficult question" is attested in general use from 1942, from the popular radio quiz show "Take It or Leave It" (debuted 1940), in which contestants answered questions of ascending difficulty and with ascending prizes for right answers, from $1 for the first to $64 for the seventh and last, and the option to take the winnings or take the next question. The show renamed itself "The $64 Question" in 1950. The amount of money offered advanced over time, and by 1955 it was "The $64,000 Question."
Baggy-browed Phil Baker took the $64 Question to Hollywood this week. As custodian of the renowned question—now so much a part of the national idiom that even $64 prose stylists avoid using it—and quizmaster of one of U.S. radio's most popular shows, Take It or Leave It (CBS, Sun., 10 p.m., E.W.T.), Phil Baker was ready to put both on celluloid. But there would be one slight variation: to suit Hollywood's philosophy, the $64 Question would become the $640 Question. [Time, March 6, 1944].
1848 as the years of a person's life between 60 and 69; by 1827 as the seventh decade of years in a given century. See sixty.
1738, "person sixty years old or between sixty and seventy years old," from Latin sexagenarius "containing sixty," from sexagenarius, from sexageni "sixty each, sixty at a time," from sexaginta "sixty," from combining form of sex (see six). With -genaria "ten times," from -ginta "tens" (from PIE *dkm-ta-, from root *dekm- "ten"). As an adjective, "pertaining to or characteristic of the age from sixty to seventy," from 1836.
"composed of or produced by sixties; pertaining to division into sixty," 1680s, from Medieval Latin sexagesimalis, from Latin sexagesimus "the sixtieth," from sexaginta "sixty." Sexagisema, "second Sunday before Lent" (eighth before Easter), is from late 14c. (Sexagesime), from Medieval Latin sexagesima (dies) "the sixtieth (day)."
"sheaves of grain placed on-end and leaning against one another in a field, arranged so as to shed rain and allow the grain to dry," early 14c., shok, perhaps from an unrecorded Old English word or from Middle Low German schok "shock of corn," originally "group of sixty," from Proto-Germanic *skukka- (source also of Old Saxon skok, Dutch schok "sixty pieces; shock of corn;" German schock "sixty," Hocke "heap of sheaves"). The original sense of this is uncertain; perhaps it is connected to the source of shock (n.1) on the notion of being "thrown" together [Century Dictionary]. The English word in 16c.-17c. sometimes was a unit of tale meaning "60-piece lot," from trade with the Dutch.