I introduce and study a nested search problem modeled as a tree structure that generalizes Weitzman (1979) in two ways: (1) search progresses incrementally, reflecting real-life scenarios where agents gradually acquire information about the prizes; and (2) the realization of prizes can be correlated, capturing similarities among them. I derive the optimal policy, which takes the form of an index solution. I apply this result to study monopolistic competition in a market with two stages of product inspection. My application illustrates that regulations on drip pricing lower equilibrium price and raise consumer surplus.