Paidea & Ludus

Paidea: Effectively, "play" for pleasure

Ludus: more constrained by rules, with a clear outcome (eg "winning")

Example 1: Grand Theft Auto IV ( although all apply IV is the one I am most familiar with)
GTA is a strong example of a game that offers both paidea and ludus. Although the player is required to complete missions to progress through the game, the player can also roam free in the city, doing as they please, although the different islands are unreachable by road vehicles until the player has progressed enough (through the missions) to unlock the seperate islands. The locking of islands is another form of Ludus along with the rules of the missions, for example when on a driving mission and you manage to destroy the car you are required to drive, you fail the mission. I think the Plaidea in this game gives the game a much longer life span.

Example 2: Left 4 Dead 2 (both apply again)
Left 4 Dead shows a very clear example of a game that offers Ludus. the objective of the game is to get from the starting point (safe room) to the end, by battling through hords of zombies using various weapons, reaching the end completes that particular mission. The game has a very linear structure, with the constricted maps and objectives guiding you through the map with pre-determined zombie filled areas along the route, until the next safe room is reached.