It all depends on context of his action and context of your expectations. Let me explain.
- If this was one of your non-romantic friends who did this, would you find it as annoying?
Consider what his question is and whether you are saying "no" because you don't want to do it, don't want to do it with him, or don't want to do it because it's him asking. I'm not saying you are wrong for saying "no" in any of those situations, just be aware of your real reasoning of your answer. He may have a different idea of why you are answering "no" and trying to get around your initial answer, thinking you'll decide it's best and "go along" with it. Whether it's actually better is entirely dependent on the situation.
Example:
Him: "Let's go to a snake farm."
You: "AAACK! No!"
Him: "But I love snakes."
You: "No!"
Him: "Don't you want to get over your fear of snakes?"
You: "Yes....."
Him: "Then let's go to a snake farm."
You: "No!"
This is pretty clear. You have a clear "knee jerk" reaction to snakes and don't want to go. You still what to get over your fear of snakes, but that's just too much right now. It may serve you better in the future to know what snakes are dangerous, which are helpful, and how to act when/if you meet one in the wild. But you're just not ready for it right then. Maybe you don't want to give him the satisfaction of seeing you freak out and constantly squirm at being around snakes.
To be clear, I'm not saying all interactions like this are clear or healthy, let alone have a benefit of learning something. But you need to figure out if the request follows this format or not.
This is crucial to figuring out if this is him being manipulative. Even if he is manipulating you, you have to figure out if it's because he wants to help you or help himself. Constantly helping himself is not good for you. Constantly helping you can be annoying, but it's coming from a better mindset.
- Is he asking in a way that he believes will be taken as humorous?
To exaggerate a question is sometimes taken as humorous. Let me reuse the last example.
Him: "Let's go to a snake farm."
You: "AAACK! No!"
Him: "Please, please, please, please, please, please, please, please?" {Puppy eyes} {Protruding lower lip}
Done occasionally, this is generally considered fine. However, over using it constantly is definitely a problem.
- Do you do this to him and how often?
Example:
You: "Let's go to the club to dance."
Him: "I'm tired, so no."
You: "Aw, com'on, we don't go anywhere anymore."
Him: "No."
You: "But I want to go dancing with you!"
Him: "No."
I'm not saying that him asking you is more, less, or equally annoying or unfair, I just want you to think about how he sees you and if he's just trying to act the same way to you as you do to him. I'm not saying this is your fault, I'm just wondering if maybe he thinks it's ok for him to do it.
However he is asking, the fact that you talked to him about it is a good first step. You say he is responding to your conversation, which is also a good first step. It takes time to change habits. I'm not going to make excuses for his actions, but he may have had reasons for thinking re-asking was ok and now needs time to unlearn this idea.
I had a co-worker that literally used the word "literally" in every sentence, literally every day for literally every thing he talked about, and sometimes literally multiple times in the same sentence, even when it literally didn't make sense to literally use it. Literally. Yes, what I just explained and demonstrated was how he talked for several weeks, until someone mentioned it to him and how annoying it was. Especially when he meant "figuratively". It took him a while to change, but he did.
Give him a chance to change permanently. If he doesn't then you have a problem. Even if he does permanently change and he still does it on occasion, try not to be mad about it. Just remind him you don't like it.
Also, is this really a "deal breaker"? Everyone has their faults. You just need to figure out if you can live with this one or not. You also need to figure out if it's manipulative and if that is actually harmful to you. This is your preference and no answer is a wrong answer, unless you can't live with the answer.
"I'm a man, but I can change, if I have to, I guess." - Red Green
https://en.wikipedia.org/wiki/The_Red_Green_Show#Conclusion_and_credits
Edit:
It may be a cultural thing. There are cultures where this may be part of their etiquette. In China, it's expected for a gift to be refused 2 or 3 times before being accepted. If you accept it right away, it's considered that you are greedy. It may also be considered rude to complain about the custom. I'm not saying this is what's going on, but it's something to consider.
Under section General Tips for Gifts-Giving in China:
Recipient often declines the gift while the giver has to inisit a bit. The recipient should not appear greedy when others give gift to him in China. So the recipient often declines the gift you give twice or for three times before accepting. You should offer the gift again after the recipient decline the gift, and he or she may finally accept your gift.
https://www.topchinatravel.com/china-guide/chinese-etiquette-gift-giving.htm
Also under section Chinese Etiquette #5 Chinese Gifts
Refuse the gift at least two times before accepting it.
https://www.taiwanese-secrets.com/chinese-etiquette/