With Aidan I had about six, I can't remember how often we washed then. We used ME Airflow for him. They dry really really quick, we still have them 7 years later . We're moving back to fitted with Callum too. So on the look out for a couple more prettier wraps.
I would say absolute minimum 4 for a newborn for three reasons (1) if bf'ing, you do have to change wraps slightly more frequently (2) with a newborn around you may not be so quick on the ball for washing and (3) using the same wraps all the time is boring! x
Last wash I was using fitteds full time and there were 4 wraps in the wash for about 2 days. But that doesn't include nights where I use wool / fleece. So I would say at least 6 for day time use, more if you're using them at night too.