They should definatly be doubeling and that fact that it didn't on that third blood draw is not a good sign. I was going through the same type of scenario where my numbers did not double, they were worried it was tubal but were finally able to see it was not by monitoring me through U/S. Eventually they saw a sac in the uterus but it wasn't growing right and after many U/S spaced apart it was obvious it was not viable and I ended up having a D&C. What made me OK with it was the fact that I was monitored and it was obvious after enough time even to me that it wasn't growing. Do what you feel comfortable with. If your Dr. is worried that it is tubal then she should have done an U/S by now, because that can be very serious business if not caught early. There is always a chance if your numbers don't double that your pregnancy will be ok, but that odds are very small. My Dr. said it is very rare. Good luck
Levels should double or close to double every 48-72 hours. If they are not it is a sign that something is not right. I would get another hcg test and an ultrasound to confirm that something is wrong before terminating. The fact that they are not dropping is a good sign.
With a normal, viable pregnancy your hCG levels should double about every 48 hours. Slow rising hCG levels suggest a pending miscarriage or possibly an ectopic pregnancy (if this is an ectopic pregnancy, waiting to terminate could adversely effect your ability to conceive in the future). I am surprised she didn't suggest an ultrasound to check where the pregnancy is.
Very rarely, if there is initially a twin pregnancy and one twin is lost, the hCG levels will fall or appear to not increase right afterward. They should return to normal quickly though. So, if that is what has happened in your case, your next hCG level should be 120 or more.