According to NOAD, the definition of home in this context is given thus:
to or at the place where one lives
Technically, it is not ungrammatical to say The children are playing home to mean The children are playing at home, since at is implicit in the definition of the adverb home. However, this form is certainly not colloquial, and I daresay no native speaker would use this in any context, formal or informal.
Here are some examples in which it is all right to drop at (informal contexts):
- He's staying home today.
- When is she usually home?
- I missed your class because I was home sick yesterday.
In the following examples, it would be incorrect to include a preposition:
- When is mom getting home?
- I want to go home!
- Son, when are you coming home?
- Are you driving home today? I need a ride.
You may notice that home is generally used alone as an adverb of place (without at, from, etc) with movement verbs, e.g. come, go, leave, stay, be, drive, dash, move, etc. Verbs of other species usually require at, e.g. dance, play, eat, relax, and whatever else one does at home!
To my ear eat home sounds all right, and I may have heard or used it myself. This may be one of a few exceptions, but it is certainly not formal. If eat home is ever used, my hunch is that eat in (which means eat at home) is falling out of favor, or, as native speakers tend to do, at is dropped for convenience and it sounds good!
- We're eating home today. (Technically, it is eat in if at is not used)
Another observation to note is in the case of non-movement verbs, the preposition may be dropped (colloquially) when the action described can be performed at home as opposed to somewhere else. Examples are few and far between, though.