Can I use singular countable nouns after any other? And are the following sentences grammatical?
1- Do you have any other book?
2- Do you know any other doctor?
3- Is there any other way to do this?
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According to Merriam-Webster, any other has two possible meanings.
The first meaning is used when we want to talk about a non-specific, single instance of the following noun. In this case, the noun is always singular, for example:
On any other day, it wouldn't have been a problem.
The second meaning relates to other instances than those that have already been mentioned. It is conventional to make the noun plural:
Do you have any other qualifications?
In your question, examples 1 and 2 relate to instances other than those already mentioned, and so a plural would sound more natural. Example 3 relates to a non-specific instance: any way of doing it would be fine, so a singular is OK.