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  1. I know that we need to use X's for belonging:

Boy's ball = the ball of the boy
Girl's book = the book of the girl

and for many:

Kids' toys = the toys of the kids
Dogs' food = the food of the dogs

But in case of "files types", should I use "files' types" or "files types"? I'm asking because the "types" are not belong to "files". "files" do not own the "types".

  1. What is the proper usage of "double belonging": the boy dogs' toys = the toys of the dogs who belong to the boy is this the correct form? or should I say: the boy's dogs' toys?
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If you have already been talking about some particular files, and now want to say something about their types, then you would use the plural possessive; something like:

When the download has completed, you should find a number of files in directory ... The files' types can be looked up in ...

And similarly if it was only one file, then "the file's type".

In any other context I would not expect a possessive, but "file" used as a qualifying noun: the file types.

For your second question, the boy's dogs' toys. (Note that in speech this is indistinguishable from the boy's dog's toys and indeed from the boys' dogs' toys. We manage perfectly well with this potential ambiguity in speech, but for some strange reason we're required to resolve the ambiguity when we write it down).

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