"Anytime" or "any time"?

A course participant recently asked me how to choose between "any time" and "anytime". Should she write "can I come over anytime?" or "can I come over any time?"

"Anytime" is an adverb - it is a word that describes a verb. So you can tell someone to come anytime or write anytime or visit anytime. In these instances "anytime" describes the preceding verbs. 

"Any time", on the other hand, is a phrase comprising an adjective and a noun. For our purposes you can just think of it as a noun - as a unit of time. You can possess a noun, so you can say, "do you have any time?" But you can't use a noun to describe a verb. 

Any meat -> Any time
If you find all that much too difficult to understand, then try this trick instead. When you come across a sentence needing "any time" or "anytime", ask yourself whether you can replace the phrase by "any meat". If so, you should use "any time". If not, you should use "anytime."

Here are a few examples to get you started:
ANY TIME
I hardly have any time to myself. (You can say, "I hardly have any meat.")
Do you have any time to visit your grandparents? (You can say, "Do you have any meat?")
Is there any time for us to go to the supermarket? (You can say, "Is there any meat?")

ANYTIME
Come visit us anytime! (You can't say, "Come visit us any meat.")
Anytime you are free, you should rest your eyes. (You can't say, "Any meat you are free.")