Allowing Greater Access to Safe and Effective Contraception Act This bill permits health savings accounts, medical savings accounts, and employer reimbursements for medications to be used for over-the-counter drugs. The bill also removes the limit on employee contributions to flexible spending accounts offered under specified health plans established pursuant to a pretax salary reduction agreement with an employer (i.e., cafeteria plans). Further, the bill requires the Food and Drug Administration to give priority review, at no cost, to applications for modifications to already-approved contraceptive drugs that are intended for routine use and available over the counter. Such drugs, however, must require a prescription for individuals under the age of 18.