I was about to ask exactly this. I did a few iterations and it converges pretty fast, nice to know someone went the extra mile to see exactly how many iterations it would take. Annoying if you were doing it by hand, but not that bad really.
About one bit (in binary equivalent) per naive iteration? Better with some linear interpolation (pseudo-Newton)?
IIRC, before e.g. continued fraction representations/approximations became available, high precision estimates of pi were based upon geometric subdivisions of a circle and computation of hundreds of terms.