I would like to know if it is grammatically correct to use may possibly to express possibility, such as in the following sentence:
Middle income countries may possibly experience a significant increase in health expenditure in the next five years.
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Although "may possibly" is one of the most common redundancies, you still have a chance of meeting it in books written by the authors whose mother tongue is English. Interestingly, the older the book, the higher your chance of meeting it is, even if it still is very slight, to say the least.
As for me, I don't think that all those authors and the editors of their books were so illiterate as not to be aware of such vocabulary error as a redundancy. So my take on it is that the use of "may possibly", denoting a rather vague, merely hypothetical possibility, sometimes doesn't seem to be so glaring a mistake as, say, reply back, repeat again, cash money, etc.
All that said, it is a redundancy, a vocabulary, not grammatical (IMO) mistake a learner should by all means avoid.