It seems most people only have two (12 week for Downs Scan and 20 week). I am getting another at 32 weeks for a low placenta, but this is only as required.
I had a few private scans, I did both the big scans private just so I could get more information and be fully talked through them. The public docs don't seem to give you as much info. I also had a few in between when I wanted to see the little monkey