Anytime means at whatever moment suits. Come round anytime, anytime you like, anytime soon.
As a one-word adverb it is American in origin and now general. Written as two words, any time is a quantity — do you have any time — and the difference is real, which is why careful writers keep them apart. Anytime is also the reply to thanks in American speech, meaning you are welcome.
1adverbanytimeAt whatever moment suits.
“Call me anytime if you need help with the move.”
anytime is an indefinite time.
Any plus time, run together. The two-word any time is the older form and still needed for the quantity sense.
Old English
ǣniġum tīman
Middle English
any time
English, today
anytime
From Middle English any time, ony tyme, from Old English ǣniġum tīman, equivalent to any + time.
Etymology from Wiktionary, CC BY-SA