Etymology
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get over (v.)
1680s, "overcome," from get (v.) + over (adv.). From 1712 as "recover from;" 1813 as "have done with."
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get (n.)
early 14c., "offspring, child," from get (v.) or beget. Meaning "what is got, booty" is from late 14c.
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get (v.)

c. 1200, from Old Norse geta (past tense gatum, past participle getenn) "to obtain, reach; to be able to; to beget; to learn; to be pleased with," a word of very broad meaning, often used almost as an auxilliary verb, also frequently in phrases (such as geta rett "to guess right"). This is from Proto-Germanic *getan (source also of Old Swedish gissa "to guess," literally "to try to get"), from PIE root *ghend- "to seize, take."

Old English, as well as Dutch and Frisian, had the verb almost exclusively in compounds (such as begietan, "to beget;" forgietan "to forget"). Vestiges of an Old English cognate *gietan remain obliquely in modern past participle gotten and original past tense gat, also Biblical begat.

In compound phrases with have and had it is grammatically redundant, but often usefully indicates possession, obligation, or necessity, or gives emphasis. The word and phrases built on it take up 29 columns in the OED 2nd edition; Century Dictionary lists seven distinct senses for to get up.

"I GOT on Horseback within ten Minutes after I received your Letter. When I GOT to Canterbury I GOT a Chaise for Town. But I GOT wet through before I GOT to Canterbury, and I HAVE GOT such a Cold as I shall not be able to GET rid of in a Hurry. I GOT to the Treasury about Noon, but first of all I GOT shaved and drest. I soon GOT into the Secret of GETTING a Memorial before the Board, but I could not GET an Answer then, however I GOT Intelligence from the Messenger that I should most likely GET one the next Morning. As soon as I GOT back to my Inn, I GOT my Supper, and GOT to Bed, it was not long before I GOT to Sleep. When I GOT up in the Morning, I GOT my Breakfast, and then GOT myself drest, that I might GET out in Time to GET an Answer to my Memorial. As soon as I GOT it, I GOT into the Chaise, and GOT to Canterbury by three: and about Tea Time, I GOT Home. I HAVE GOT No thing particular for you, and so Adieu." [Philip Withers, "Aristarchus, or the Principles of Composition," London, 1789, illustrating the widespread use of the verb in Modern English]

As a command to "go, be off" by 1864, American English. Meaning "to seize mentally, grasp" is from 1892. Get wind of "become acquainted with" is from 1840, from earlier to get wind "to get out, become known" (1722). To get drunk is from 1660s; to get religion is from 1772; to get better "recover health" is from 1776. To get ready "prepare oneself" is from 1890; to get going "begin, start doing something" is by 1869 in American English; get busy "go into action, begin operation" is from 1904. Get lost as a command to go away is by 1947. To get ahead "make progress" is from 1807. To get to (someone) "vex, fret, obsess" is by 1961, American English (get alone as "to puzzle, trouble, annoy" is by 1867, American English). To get out of hand originally (1765) meant "to advance beyond the need for guidance;" sense of "to break free, run wild" is from 1892, from horsemanship. To get on (someone's) nerves is attested by 1970.

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over- 

word-forming element meaning variously "above; highest; across; higher in power or authority; too much; above normal; outer; beyond in time, too long," from Old English ofer (from PIE root *uper "over"). Over and its Germanic relations were widely used as prefixes, and sometimes could be used with negative force. This is rare in Modern English, but compare Gothic ufarmunnon "to forget," ufar-swaran "to swear falsely;" Old English ofercræft "fraud."

In some of its uses, moreover, over is a movable element, which can be prefixed at will to almost any verb or adjective of suitable sense, as freely as an adjective can be placed before a substantive or an adverb before an adjective. [OED]

Among the old words not now existing are Old English oferlufu (Middle English oferlufe), literally "over-love," hence "excessive or immoderate love." Over- in Middle English also could carry a sense of "too little, below normal," as in over-lyght "of too little weight" (c. 1400), overlitel "too small" (mid-14c.), overshort, etc.

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over (prep., adv.)

Old English ofer "beyond; above, in place or position higher than; upon; in; across, past; more than; on high," from Proto-Germanic *uberi (source also of Old Saxon obar, Old Frisian over, Old Norse yfir, Old High German ubar, German über, Gothic ufar "over, above"), from PIE root *uper "over."

As an adjective from Old English uffera. The senses of "past, done,  finished; through the whole extent, from beginning to end" are attested from late 14c. The sense of "so as to cover the whole surface" is from c. 1400. Meaning "leaning forward and down" is from 1540s. The meaning "recovered from" is from 1929. In radio communication, it is used to indicate the speaker has finished speaking (1926).

Above expresses greater elevation, but not necessarily in or near a perpendicular direction; over expresses perpendicularity or something near it: thus, one cloud may be above another, without being over it. Over often implies motion or extension where above would not; hence the difference in sense of the flying of a bird over or above a house, the hanging of a branch over or above a wall. In such uses over seems to represent greater nearness. [Century Dictionary]

Phrase over and above (mid-15c.) is pleonastic, for emphasis. Adjective phrase over-the-counter is attested from 1875, originally of stocks and shares. To be (someone) all over "be exactly what one expects of (someone)" is by 1721.

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get-away (n.)
also getaway, 1852, "an escape," originally in fox hunting, from verbal phrase get away "escape" (early 14c.); see get (v.) + away (adv.). Of prisoners or criminals from 1893.
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fly-over (n.)
also flyover, 1901 of bridges, 1931, of aircraft flights, from fly (v.1) + over (adv.).
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sleep-over (n.)
1935, from verbal phrase; see sleep (v.) + over (adv.).
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walk-over (n.)
"easy victory," 1838, such as one that happens in the absence of competitors, when the solitary starter, being obliged to complete the event, can traverse the course at a walk. Transferred sense of "anything accomplished with great ease" is attested from 1902. To walk (all) over (someone) "treat with contempt" is from 1851.
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cross-over (n.)

also crossover, 1795, a term in calico-printing, "superimposed color in the form of stripes or crossbars," from the verbal phrase; see cross (v.) + over (adv.). From 1884 in railroading; from 1912 in biology. As a general adjective from 1893; specifically of musicians and genres from 1971.

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