c. 1200, "latest, final, following all others," a contraction of Old English latost (adj.) "slowest, latest," superlative of læt (see late); in some uses from late (adv.). Cognate with Old Frisian lest, Dutch laatst, Old High German laggost, German letzt.
Meaning "last in space, furthest, most remote" is from late 14c.; meaning "most unlikely or unsuitable" is from mid-15c. Meaning "most recent, next before the present" (as in last night, last September) is from late 14c.; latest would be more correct, but idiom rules and the last time I saw her might mean the most recent time this hour or the final time forever.
The biblical last days ("belonging to the end") is attested from late 14c. Last hurrah is from the title of Edwin O'Connor's 1956 novel. Last word "final, definitive statement" is from 1650s. A dying person's last words so called by 1740. As an adjective, last-minute attested from 1913. Last-chance (adj.) is from 1962. Expression if it's the last thing I do, expressing strong determination, is attested by 1905.
"wooden model of a human foot used by shoemakers," from Old English læste "shoemaker's last," earlier last "track, footprint, footstep, trace," from Proto-Germanic *laisti- (source also of Old Norse leistr "the foot," Middle Dutch, Dutch leest "form, model, last," Old High German leist "track, footprint," German Leisten "last," Gothic laistjan "to follow"), related to Old English læran "to teach," from PIE root *lois- "furrow, track."
"endure, go on existing," from Old English læstan "to continue, endure," earlier "follow (a leader), accomplish, carry out, perform," from Proto-Germanic *laistjan "to follow a track" (source also of Gothic laistjan "to follow after," Old Frisian lasta "to fulfill, to pay (duties)," German leisten "to perform, achieve, afford"), from PIE root *lois- "furrow, track." It is related to last (n.1), but not to last (adj.). Related: Lasted; lasting.
c. 1300 as an adverbial phrase, "completely, thoroughly, to the utmost degree," from out (adv.). Adjective usage is attested by 1813.
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 utian "expel, put out," from the source of out (adv.). It has been used in many specific senses over the years; the meaning "disclose to public view, reveal, make known" is by mid-14c.
Eufrosyne preyde Þat god schulde not outen hire to nowiht. ["Legendary of St. Euphrosyne," c. 1350]
Meaning "to expose as a closet homosexual" is first by 1990 (as an adjective meaning "openly avowing one's homosexuality" it dates from 1970s; see closet). To come out "declare oneself publicly as homosexual" is from 1968 and probably short for come out of the closet. Related: Outed; outing. Compare outen.
late Old English, "outer," from out (adv.). From mid-13c. as "that is or lies on the outside, exterior." Of a light or candle, "extinguished, no longer burning," c. 1300. Sense of "no longer secret" is by 1713. Sense in baseball (1860) was earlier in cricket (1746). Meaning "unconscious" is attested from 1898, originally in boxing from the notion of "defeated ('out') by failing to rise within a 10-count." To be out on one's feet is from 1952. From 1966 as "unfashionable, not stylish, popular, or modern."