"He asks me to stop."
Translation:Hij verzoekt mij om te stoppen.
Not always! "Verzoeken" specifically means to request, to ask someone to perform an action. "Vragen" can also be used in that way, but is more generally used for asking questions.
so not 'om me te stoppen'? I've seen that order in other phrases like 'om kaas te verkopen'.
(I think) in the question, mij is an indirect object. It is not the subject of what needs to be stopped, but the subject of the verb verzoekt.
As this sentence does not use the preposition aan, we must place mij (or perhaps me instead, but don't quote me on that) right after the verb (in this case, verzoekt). Here's an e.g. with aan (in the Present Continuous):
Hij bent om te stoppen aan mij verzoeken (note that mij is now at the other end of the sentence)
I hope I was able to help. I'm only beginning to learn how to put as much blood, sweat, and tears (oh, so many, many tears) into other DLers' educative (is that even a word?!) experience as I do into my own.