After the first! If after they say no, then later they change their mind, they'll let you know. Move on, continuing to ask will only make them uncomfortable and feel pressured, and even afraid, even if you mean no harm. Chances are after the first rejection you could still possibly be friends, and that after the thirtieth they'll be calling the cops. Anyways continuing to ask isn't even healthy for yourself! Even if you're hung up on this person, try to move on, and if you can't, being friends is better than nothing right?
Never. You've got a bit of thinking to do though. If your getting knocked back all the time then you're fucking up somewhere. You should approach a girl like you already pulled her. Walk over like you own the place and just talk. Keep a smile on your face and look in her eyes when your talking. Women react to body language and are harsh critics. You're goal is not to bore her with your shit. Let her bore you with hers. Everybody loves talking about themselves and since most women are notoriously introverted they'll love you asking personal questions.
After 3. The first time could be a surprise. The second time is more like a wild card. No girl rejects a guy she likes three times in a row. If you keep trying after 3, the person might take advantage of you or might label you creepy. Stop with 3.
I'd stick with the rule of 3's, but what's more important is when you try again, and I suppose the circumstances are different for everyone, and so is the girl and her tolerance. Just try not to come off desperate!
If she's not into you and she's saying a straight up no. Then that means stop after 1 but if she's being kind of flirty or nervous about it you can try a couple more times but no more than 3 other wise you could seem desperate
After the 2nd try. A man should never disrespect himself by crawling around for a woman. If she doesn't want you the first time, cool, maybe she didn't think about it. If at the second time she still rejects you, move on, she doesn't want you, there are millions of women out there, go ask them.
Depends if you mean a stranger, someone you know but never dated or an ex. Tenacity might be. Useful tool depending on the scenario. But there are a lot of variables so there's not really a specific number to give. For the most part I'd say if she gives you a stern no, then once is more than enough to back off.