First of all, if a girl tells you that, I figure it's nothing like flipping a coin with 99% heads, 1% tails, and stopping when you get tails.
However, if we accept that model, it's a well-defined problem with a clear answer. The expected number of times you have to ask is just the average number of times you would have to ask if you repeated this process many times. (It's possible you would be really lucky, and she'd agree the first time; it's possible you would be really unlucky, and she'd turn you down 200 times before accepting.) We can apply the same kind of reasoning as in this question. (If people keep having children until they get a boy, how many children will the average family end up with?)
Suppose 1 million guys ask this girl out again and again until she says yes, and her responses really are random, with just a 1% chance of acceptance each time. The 1% chance of acceptance means that about 1 out of every 100 responses was a yes. So once all 1 million guys have gotten their dates, we have 1 million yes responses and about 99 million no responses.
So on average, each guy had to ask 100 times.
See the answer to the linked question for the infinite sum to obtain the same answer.