For me the finest moments in games are about discovery.
A discovery-based game will present the player with a motivation, but not a specific objective. The most interesting puzzle is to define the objective, to explore and discover the question you are supposed to find the answer to. To discover the world and the rules without any handholding.
Games that do this include Star Control 2 (my favorite game of all time), Divine Divinity, The Last Express, Antichamber, Starseed Pilgrim, Portal, Spelunky, Project Nomad, and the upcoming games The Witness and Divinity: Original Sin.
These are higher order puzzles, as opposed to limited, read-the-designers-mind inventory puzzles.
Adventure games rarely do this—exception The Last Express—but they sometimes simulate it by creating a sense of discovery of a fully realized and fascinating world of wonder, and by allowing the player to visit a few different locations on a map and figure out what he needs to do next.
The best examples of this are probably The Longest Journey and GK1.