What happens when you buy all the landmarks?

Landmarks are not purchased, you just renovate them until they are complete and move to the next one in the following order: lighthouse, chapel, pirate ship, mountain chateau, and aviary. Once you complete the aviary you continue to earn flowers by hatching birds and releasing them on your island.

If you are referring to 3x3 buildings, there are currently 26 of them to buy. There is rumor that there may not be any more to come.