"any longer, to any extent," one-word form by 1865, from any + more. Typically used with a negative, a custom as old as Middle English, where without any more is found late 14c.
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any adj., pron.
"one, a or an, some," Old English ænig (adjective, pronoun) "any, anyone," literally "one-y," from Proto-Germanic *ainagas (source also of Old Saxon enig, Old Norse einigr, Old Frisian enich, Dutch enig, German einig), from PIE root *oi-no- "one, unique." The -y may have diminutive force here.
As a noun, late 12c.; as an adverb, "in any degree," c. 1400. Emphatic form any old______ (British variant: any bloody ______) is recorded from 1896. At any rate is recorded from 1847. Among the large family of compounds beginning with any-, anykyn "any kind" (c. 1300) did not survive, and Anywhen (1831) is rarely used, but OED calls it "common in Southern [English] dialects."
[A]ani refers to single entities, amounts, etc., occurring at random or chosen at random, as being convenient, suitable, to one's liking, etc. It is frequently emphatic and generalizing, having the force of 'any whatever, any at all' and 'any and every'. It is common in questions, conditional clauses, and negative statements, but not in affirmative statements (where som is used instead). [The Middle English Compendium]
more adj.
Old English mara "greater, relatively greater, more, stronger, mightier," used as a comparative of micel "great" (see mickle), from Proto-Germanic *maiz (source also of Old Saxon mera, Old Norse meiri, Old Frisian mara, Middle Dutch mere, Old High German meriro, German mehr, Gothic maiza), from PIE *meis- (source also of Avestan mazja "greater," Old Irish mor "great," Welsh mawr "great," Greek -moros "great," Oscan mais "more"), perhaps from a root *me- "big."
Sometimes used as an adverb in Old English ("in addition"), but Old English generally used related ma "more" as adverb and noun. This became Middle English mo, but more in this sense began to predominate in later Middle English.
"Take some more tea," the March Hare said to Alice, very earnestly.
"I've had nothing yet," Alice replied in an offended tone, "so I can't take more."
"You mean you can't take less," said the Hatter: "it's very easy to take more than nothing."
As a noun, "a greater quantity, amount, or number," in Old English. More and more "larger and larger amounts" is from 12c. More or less "in a greater or lesser degree" is from early 13c.; appended to a statement to indicate nearness but not precision, from 1580s. The more the merrier "the larger the company the greater the enjoyment" is from late 14c. (þe mo þe myryer).