Etymology
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incorrigible (adj.)
mid-14c., "incurable (of diseases, venom, etc.); extravagant (of expense); implacable (of hearts)," from Old French incorrigible "perfect, beyond rebuke or discipline" (14c.) or directly from Latin incorrigibilis "not to be corrected," from in- "not, opposite of" (see in- (1)) + corrigibilis, from corrigere "to correct," from com-, intensive prefix (see com-), + regere "to lead straight, rule" (from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule"). From mid-15c. as "incapable of improvement" (of persons). Related: Incorrigibly. As a noun, from 1746.
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incorrigibility (n.)

"incapability of correction or amendment," late 15c., incorrigibilite, from Medieval Latin incorrigibilitas; see incorrigible + -ity.

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*reg- 

Proto-Indo-European root meaning "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule."

It forms all or part of: abrogate; address; adroit; Alaric; alert; anorectic; anorexia; arrogant; arrogate; bishopric; correct; corvee; derecho; derogate; derogatory; Dietrich; direct; dress; eldritch; erect; ergo; Eric; Frederick; Henry; incorrigible; interregnum; interrogate; maharajah; Maratha; prerogative; prorogue; rack (n.1) "frame with bars;" rail (n.1) "horizontal bar passing from one post or support to another;" Raj; rajah; rake (n.1) "toothed tool for drawing or scraping things together;" rake (n.2) "debauchee; idle, dissolute person;" rakish; rank (adj.) "corrupt, loathsome, foul;" real (n.) "small Spanish silver coin;" realm; reck; reckless; reckon; rectangle; rectify; rectilinear; rectitude; recto; recto-; rector; rectum; regal; regent; regicide; regime; regimen; regiment; region; regular; regulate; Regulus; Reich; reign; resurgent; rex; rich; right; Risorgimento; rogation; royal; rule; sord; source; subrogate; subrogation; surge; surrogate; viceroy.

It is the hypothetical source of/evidence for its existence is provided by:

Sanskrit raj- "a king, a leader," rjyati "he stretches himself," riag "torture" (by racking); Avestan razeyeiti "directs," raštva- "directed, arranged, straight;" Persian rahst "right, correct;" Latin regere "to rule, direct, lead, govern," rex (genitive regis) "king," rectus "right, correct;" Greek oregein "to reach, extend;" Old Irish ri, Gaelic righ "a king," Gaulish -rix "a king" (in personal names, such as Vircingetorix), Old Irish rigim "to stretch out;" Gothic reiks "a leader," raihts "straight, right;" Lithuanian raižytis "to stretch oneself;" Old English rice "kingdom," -ric "king," rice "rich, powerful," riht "correct;" Gothic raihts, Old High German recht, Old Swedish reht, Old Norse rettr "correct."

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hopeful (adj.)
c. 1200, "full of hope," from hope (n.) + -ful. From 1560s as "having qualities which excited hope." As a noun, "one on whom hopes are set," from 1720. Often ironic in colloquial use, of willful or incorrigible offspring. Related: Hopefulness.
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noob (n.)

c. 2000 in gamer slang, variant of newbie; often used interchangeably with it, but also often with a more derogatory shade of meaning; newbies owe their clueless behavior to lack of experience and are capable of improvement, while the fundamental characteristic of noobs is incorrigible obnoxiousness or stupidity.

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cucking-stool (n.)
The Cucking-stool was a means adopted for the punishment of scolds and incorrigible women by ducking them in the water, after having secured them in a chair or stool, fixed at the end of a long pole, serving as a lever by which they were immersed in some muddy or stinking pond. [Willis's Current Notes, February 1851]

early 13c., from verbal noun from cuck "to void excrement," from Old Norse kuka "feces," from PIE root *kakka- "to defecate." So called because they sometimes resembled the old close stool of the pre-plumbing days, a portable indoor toilet that looked like a chair with a box under the seat. Old folk etymology made the first element a corruption of cotquean. For second element, see stool. Also known as trebucket and castigatory, it was used on fraudulent tradesmen, in addition to disorderly women, either for public exposure to ridicule or for ducking.

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