What do I say for somebody keeps bringing up and talking into somebody's ear about the same topic again and again till he is fed up and finally agrees to it?
Example: Mary's been [word or idiom] to John till he finally bought it for her.
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It's called nagging if it's really annoying.
Best use the simple past tense though if the subordinate sentence is in the past simple.
Mary nagged at John till he finally succumbed and bought it for her.
She's been nagging about it since this morning
He's always nagging at her for staying out late
From OALD the phrasal verb to harp on about sth is one possibility, but it is informal.
This other one, the phrasal verb to bang on (aboutsth), used in British English only, (OALD) is informal too.
Pester (verb): If you say that someone is pestering you, you mean that they keep asking you to do something, or keep talking to you, and you find this annoying — Collins Dictionary
Example: John has been pestering her to go out with him all month.
Alternatively, you could say Persecute, badger or plague.