sixty-nine adj., n.

"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.

"the cardinal number one more than eight or one less than ten; the number which is one more than eight;" Middle English nīn, from Old English nigen, from Proto-Germanic *newun (source also of Old Saxon nigun, Old Frisian niugun, Old Norse niu, Swedish nio, Middle Dutch neghen, Dutch negen, Old High German niun, German neun, Gothic niun "nine").

This is from PIE root *newn "nine" (source also of Sanskrit nava, Avestan nava, Greek ennea (with unetymological initial e-), Albanian nende, Latin novem (with change of -n- to -m- by analogy of septem, decem), Lithuanian devyni, Old Church Slavonic deveti (the Balto-Slavic forms by dissimilation of -n- to -d-), Old Irish noin, Welsh naw).

As "a symbol representing the number nine," late 14c. The proverbial nine lives of a cat are attested from 17c.  Nine-to-five "the average workday" is attested from 1935. Nine days (or nights) has been proverbial since mid-14c. for the time which a wonder or novelty holds attention; the expression nine days' wonder is from 1590s. The Nine "the Muses" is from c. 1600. Also see nines.