There was a time in the past when the only acceptable to write any time is if it would be written in two words. It is only recently when anytime started to be written as one word. If you are in doubt about the right term to use, it is more grammatically correct to use any time instead of any time.
You cannot use anytime with the preposition at. If in case you truly need to use the preposition, you always have the use the two-word version in order to avoid any confusion. If you are also going to speak about the amount of time that you have used up, or the time used to finish a project, you still need to use the two-word version.
While there may not seem to be any differences between the words anytime and any time, there are differences which can segregate the words from one another. Any time and anytime are explained in Webster's Dictionary as the same thing, and they mean at any point in time or whenever. Britain's use of the word influenced the separation of the two words.
The correct way to use any time is when the word 'at' precedes 'at any time.' The word 'at' is said to generate a thorough separation between the two words. For example, 'You can come by at any time.' Anytime is the preferred in American English because it is the quicker use of the word. Both any time and anytime can be located in the dictionary after the words anything and anyway.