last (adj.) Look up last at Dictionary.com
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 "most recent, next before the present" (last month) is from late 14c., as is that of "last in space, furthest, most remote." The biblical last days ("belonging to the end") is attested from late 14c. Meaning "most unlikely or unsuitable" is from mid-15c.

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.
last (v.) Look up last at Dictionary.com
"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 *leis- (1) "track, furrow" (see learn). Related to last (n.1), not to last (adj.). Related: Lasted; lasting.
last (n.1) Look up last at Dictionary.com
"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 *laist- (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," Old English læran "to teach"), from PIE root *leis- (1) "track, furrow" (see learn). Related to last (v.).
last (n.2) Look up last at Dictionary.com
late Old English, "the last or final man, object, time, etc.," from last (adj.). From late 14c. as "most recent person, latest comer." Also in Middle English as a noun, "duration" (early 14c.), from the verb. Phrase at (the) last is from c. 1200; extended form long last is from 1520s. To the last is from c. 1400.
last (adv.) Look up last at Dictionary.com
c. 1200, "most recently;" early 13c., "finally, after all others" (contrasted to first), contraction of Old English lætest (adv.), superlative of late (see late).