Why would a person want to take a continuous or extended dose birth control pill?

To avoid menses: The 'period' you have on birth control pills or NuvaRing is a chemical period, because the Progesterone and estrogen are removed. It is not a 'real' period. The Progesterone does not allow the lining to thicken, hence preventing uterine cancer. There is no reason to have a period on birth control pills, unless they are specifically used for cycle control.

Suppress menses: Menstruation is not physiologically necessary. Suppression of menstruation has medical and personal benefits for many women, and is cost-effective. For women whose goal is reduction, but not necessarily elimination, of monthly blood flow, cyclic use of estrogen-progestin contraceptives can be used. Some of the medical benefits include reduction in risk of ovarian and endometrial cancer.