I love open-ended games.  I love wandering off the beaten path to find things nobody's heard of, or at least, things the NPCs didn't tell me to go find.  These games know what I'm talkin' about, and they deliver.

Spore

Spore takes open-ended-ness to a new level.  By letting you customize the creature's powers and traits, even the same areas become different on a replay, but forget all that.  Right from the earliest levels of creature creation, you never know what's off the horizon of the screen – fertile feeding grounds, or a huge predator eating everything in sight. 

GTA

Every game in the Grand Theft Auto series has been amazingly open-ended.  Nevermind that there might only be one endgame mission path, the missions are just a guide to playing in the detailed urban world.  NPCs are quirky – walking around on foot observing characters leads to all kinds of encounters that can be lucrative or just make you say “wow – what?!”  The open-ended design of Grand Theft Auto games has only increased with San Andreas and GTA IV.

Katamari

Katamari Damacy, We Love Katamari, My & My Katamari and Beautiful Katamari – four games that not only focus on providing huge freedom in how you approach each map, but of course, they also give you the scale factor of traveling around in inch-sized or mile-sized obstacle courses.  No other game series lets you start by collecting paperclips and knickknacks in an office building, then after some more exploring, come back and grab the building itself later.

The Sims

Nothing makes a situation more random and interesting than people, and this game series, simulating people who deal with all the different factors of living, is definitely both interesting and random.  Secret crushes!  Stupid stubborn grudges!  Food cravings!  The perils of chronic laziness!  The urge to go plant a garden instead of going to work!  All kinds of perfectly normal human urges manifest in the curious minds of these Sims, and the result is really a unique story each time you play it, made better by the fact that there's no cut-scenes, just living-as-gameplay.