1. A notification system that provides an electronic notification to an intended recipient, the notification system comprising an online service provider system including at least one processing device and including: a notification service configured to enable an originator of an electronic notification to input attribute information for the electronic notification that includes: future time information accessible as a basis for determining a future time for making a plurality of electronic messages that each include the electronic notification accessible to the intended recipient, the plurality of electronic messages including a particular electronic message and at least one other electronic message, and expiration time information accessible as a basis for determining an expiration time subsequent to the future time for making the plurality of electronic messages inaccessible to the intended recipient; and a notification delivery service configured to: determine the future time based on the future time information, determine the expiration time based on tile expiration time information, make a plurality of electronic messages accessible to the intended recipient subsequent to the future time but prior to the expiration time, determine whether the intended recipient accesses the accessible particular electronic message prior to the expiration time, and remove accessibility by the intended recipient to the at least one other electronic message while concurrently maintaining accessibility by the intended recipient to the particular electronic message conditioned on determining that the intended recipient has accessed the particular electronic message prior to the expiration time.