"anymore" vs. "any more"

Solution 1:

anymore is used, when you compare the status quo to the status quo ante: Something that was does no longer apply to the status quo, but it was thus earlier.

For instance:

I can't take it anymore.

Previously, you didn't very much or it just didn't bother you. But now, it has become too much to bear, and you cannot continue like that any longer.

any more are two words, where you can take the meaning of each word quite literal and it refers to more of something than you currently have/want.

In a question:

Are there any more surprises

You would be wondering, if there are additional surprises - thereby implying that there were already some.

anymore requests

I can't think of an example where this would be grammatically correct.

any more requests

This is correct.

Solution 2:

As this NGram shows, the single-word form is a relatively recent innovation...

If in doubt, put the space in. I can't think of any sentence where that would actually be considered incorrect by anyone. But as OP indicates, there are definitely contexts where most if not all native speakers would object to the single-word form.

I don't really think it makes much sense to claim different meanings for the two forms, but here's someone making a good stab at it.

EDIT: (Many years later! :)

I just found this site, which says Anymore means at the present time, any longer, or from now on, whereas Any more means anything or something additional or further.

They also say Anymore is not normally considered an actual word in UK English, which explains my earlier position. But older and wiser, I can see the point of the distinction...

1:) I have enough staff. I don't need to hire anymore. (continued hiring is unnecessary)
2:) I have enough staff. I don't need to hire any more. (more staff are unnecessary)

Solution 3:

I wonder if it could be a US x UK issue. I have always learnt and used 'any more' (two words) in all contexts, and come from a British family unit, attending a British School. So we would have: John doesn't live here any more (UK) John doesn't live here anymore (US)

Solution 4:

They are very similar, but there is one difference:

Are there any more guests coming? The boat can't stand the weight anymore.

Any more is used to mean "additional", while anymore means 'any longer'

As you can see, there are instances when any more needs to be used to clarify from anymore.

Solution 5:

You are right, the second example, "anymore requests" is incorrect because anymore is an adverb. According to Dictionary.com, anymore carries a negative connotation, too, and also can mean "nowadays" in a non-negative way. I have heard it used that way here in SE USA.

For the negative connotation see also: http://www.youtube.com/watch?v=THtX7H6ZJi8