Any more means still more, as in do you have any more cake? Anymore means any longer, as in do you have cake anymore? They are different phrases with different meanings and slightly different pronunciation. In particular, any more has two lexical stresses whereas anymore has only one. Any more refers to a quantity of some other thing whereas anymore is always about time.
This is the distinction in American English at any rate. Perhaps they write "do you have cake any more" in British English, but it certainly looks wrong to me, and I doubt they pronounce it in this phrase with two lexical stresses. So anymore may be incorrect according to British rules of orthography, but then the British pronounce lieutenant with an eff because some posh person misread the u in that as a v and everyone else has been afraid since then to pronounce it otherwise lest they sound uneducated, so take that as you will. Americans and British hypercorrect their English in different ways. Vive la différence!