When he asked for the third time, the angry policeman told him again of Mr. Lane’s defeat and retirement.
Why not use ‘the third time’ instead of ‘for the third time’? I can not tell the difference between these two phrases.
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The preposition "for" is used to indicate the object, aim, or purpose of an action or activity. In other words, it makes what follows (ie "the third time") noteworthy, rather than just an aside.
The difference is better illustrated by switching it to "the last time".
When he asked the last time...
This would refer to the last, or most recent time he asked, but doesn't exclude the possibility that he may ask again in the future.
When he asked for the last time.
This would mean that the occasion referred to was the last, or final time, and he did not ask again after that. This is because the preposition makes "the last time" significant - you're making a point of saying that it was the last, not just the latest.
In your examples, the difference may seem less stark. Basically, adding "for" makes it more significant that it was the third time... perhaps the person asking was counting the number of times he asked, or was simply more aware of how many times he had asked? Or, perhaps there was some other significance to him asking three times that achieved a particular result? Omitting the preposition makes the number of asks just an aside - maybe he wasn't counting, maybe there was nothing special about it being the third time, and what followed just happened to be the third time.