Not Microsoft Word this time, but a similar spelling/grammar checker. I typed “to the best of my ability” and it blue-underlined my, suggesting me: “to the best of me ability”. No, no, no. Certainly not in formal writing (though I note that Pages for Mac and WordPress don’t question me ability (or, less surprisingly, my ability). Just maybe in very informal, non-standard speech, by some people.
I can find very little information about this usage, probably because it is so informal. This inconclusive ELL Stack Exchange discussion is the only one so far. It’s probably a variant pronunciation of my rather than actually me. People who say to the best of me ability don’t say to the best of you/he/she/it/us/they ability instead of your/his/her/its/our/their ability (the only possible pronunciation is ya ability). Compare I’ll do my best, I’ll do m’best and I’ll do me best with to the best of my ability, *to the best of m’ability and ?to the best of me ability. Note also that me in this usage can’t be stressed: Me car’s been stolen! v Not your car, me car!
All of Google Ngrams’ results for me *_NOUN are from the bigger construction V me N; for example, me something from tell/give/show/teach me something.
I wouldn’t be able to program a spelling/grammar check, so maybe I shouldn’t criticise, but I ever did, I wouldn’t question my N (unless is was part of a V me something construction).