1911 as one word; two-word form is in Middle English (late 14c.); from any + place (n.).
Entries linking to anyplace
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]
place n.
c. 1200, "space, dimensional extent, room, area," from Old French place "place, spot" (12c.) and directly from Medieval Latin placea "place, spot," from Latin platea "courtyard, open space; broad way, avenue," from Greek plateia (hodos) "broad (way)," fem. of platys "broad," from PIE root *plat- "to spread."
Replaced Old English stow and stede. From mid-13c. as "particular part of space, extent, definite location, spot, site;" from early 14c. as "position or place occupied by custom, etc.; precedence, priority in rank or dignity; social status, position on some social scale;" from late 14c. as "inhabited place, town, country," also "place on the surface of something, portion of something, part." Meaning "a situation, appointment, or employment" is by 1550s. Meaning "group of houses in a town" is from 1580s.
Also from the same Latin source are Italian piazza, Catalan plassa, Spanish plaza, Middle Dutch plaetse, Dutch plaats, German Platz, Danish plads, Norwegian plass. The word appears via the Bible in Old English (Old Northumbrian plaece, plaetse "an open place in a city"), but the modern word is a reborrowing.
Sense of "a mansion with its adjoining grounds" is from mid-14c.; that of "building or part of a building set apart for some purpose is by late 15c. (in place of worship). Meaning "a broad way, square, or open space in a city or town," often having some particular use or character (Park Place, Waverly Place,Rillington Place) is by 1690s, from a sense in French. Its wide application in English covers meanings that in French require three words: place, lieu, and endroit. Cognate Italian piazza and Spanish plaza retain more of the etymological sense.
To take place "happen, come to pass, be accomplished" (mid-15c., earlier have place, late 14c.), translates French avoir lieu. To know (one's) place "know how to behave in a manner befitting one's rank, situation, etc." is from c. 1600, from the "social status" sense; hence the figurative expression put (someone) in his or her place (1855). In in the first place, etc., it has the sense of "point or degree in order of proceeding" (1630s). Out of place "not properly adjusted or placed in relation to other things" is by 1520s. All over the place "in disorder" is attested from 1923.