Etymology
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at- 
assimilated form of ad- "to, toward, before" before stems beginning in -t-; see ad-. In Old French and Middle English regularly reduced to a-, later restored.
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at (prep.)

Old English æt, from Proto-Germanic *at (source also of Old Norse, Gothic at, Old Frisian et, Old High German az), from PIE root *ad- "to, near, at." Lost in German and Dutch, which use their equivalent of to; in Scandinavian, however, to has been lost and at fills its place.

At is used to denote relations of so many kinds, and some of these so remote from its primary local sense, that a classification of its uses is very difficult. [OED]

In choosing between at church, in church, etc. at is properly distinguished from in or on by involving some practical connection; a worshipper is at church; a tourist is in the church. In 19c. it was used for points of the compass as regions of a country (at the South) where later tendency was to use in.

The colloquial use of at after where (as in where it's at) is noted in Bartlett (1859). At last is recorded from late 13c.; adverbial phrase at least was in use by 1775. At in Middle English was used freely with prepositions (as in at after, which is in Shakespeare), but this has faded with the exception of at about.

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at all (prep.)
"in any way," mid-14c., originally used only affirmatively (as in I Samuel xx.6 in KJV: "If thy father at all misse me"); now it is overwhelmingly used only in the negative or in interrogatory expressions, formerly also in literary attempts at Irish dialect.
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at bay (prep.)
late 14c., originally often at the bay; see bay (n.3). Figurative use, of human beings in difficulties, is from c. 1400. The expression reflects the former more widespread use of at. The earlier form of the phrase was at abai, used of hunted animals, "unable to escape," c. 1300, from French.
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at-bat (n.)
"baseball player's turn at the plate," 1912, originally a column heading in statistics tables, from the prepositional phrase.
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at-home (n.)
"reception of visitors," 1745, noun use of prepositional phrase at home.
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come-at-able (adj.)

"capable of being approached, attainable," 1680s, from come + at + -able.

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stay-at-home (n.)
1836, from adjectival stay-at-home (1806); see stay (v.1).
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batty (adj.)
1580s, "pertaining to or resembling a bat or bats," from bat (n.2) + -y (2). Slang sense "nuts, crazy" is attested from 1903, from the colloquial expression (to have) bats in (one's) belfry "not be right in the head" (1899).
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batboy (n.)
also bat-boy, 1910, "youth who has charge of the bats and other equipment of a baseball team," from bat (n.1) + boy.
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