Typically only one, unless the doctor has a reason to do another. I ended up having two ultrasounds each pregnancy, for various reasons, and didn’t have any trouble getting my insurance to pay for them. Normally, you have an ultrasound around 16 weeks to check the size of the baby, look for any potential problems, and learn the sex of the baby if you choose. If everything goes very smoothly, that’s the only ultrasound the doctor will do.
My “extra” ultrasounds were for light bleeding, to check the position of the placenta, and to check the size again of one of my boys, who was rather large. In each case, the doctor suggested the ultrasounds and it wasn’t something I asked for.
I believe there are private ultrasound companies that you can pay for an additional ultrasound. It’s more like taking your baby to get his or her pictures taken—there’s no medical advantage, but they give you a video.