There's no time limit. No restraints, other than the occasional Insurmountable Waist-Height Fence. No objectives, no requirements. You can do whatever you want, whenever you want. Well, okay, there are goals if you actually want to, you know, finish the game, but why bother when there's so much to do beforehand?
As computer technology evolves, video games gradually move away from the traditional sequential level design towards letting their players loose onto a vast virtual world devoid of designated transition choke-points and loading screens. The terms "open world" and "sandbox" are applied interchangeably to such design, but they are actually two separate aspects of a Wide Open Sandbox that have to be distinguished:
Sandbox gameplay is a type of gameplay that offers a large variety of mechanics and game systems for the player to explore and toy with, from Sidequests to Mini Games. This is in contrast to games revolving around a few core gameplay mechanics (like Portal).
Although these two aspects often go hand in hand, there is a large number of games that have open worlds but no sandbox gameplay (e.g. the Metroidvania genre). Conversely, there are sandbox games with a wide variety of gameplay systems but little effective freedom of movement (e.g. the "survival sandbox" subgenre in the vein of Minecraft). Only when the two aspects come together in a single game does it count as a proper Wide Open Sandbox.
The appeal of an open-world sandbox lies in its massive Replay Value potential, as the players are theoretically free to set and pursue their own goals and experiment with various gameplay systems indefinitely. The flip side is that many developers fall (and lead their players) into the Quicksand Box trap, where the sheer wealth of non-linear content, without a driving storyline or a clear overarching goal, results in players losing interest in the game, especially if it doesn't keep track of their objectives and progress. Additionally, player freedom is often difficult to reconcile with a dramatically paced narrative, resulting in underwhelming plots and characterization in many open-world sandbox games.
Historically, open-world sandboxes took many cues from role-playing video games, especially of the Western variety, which have traditionally implemented an assortment of subsystems and are often driven by exploration. An RPG player, however, is rarely expected to be Sidetracked by the Gold Saucer, which is the bread-and-butter of sandbox games. MMORPGs and MUDs tend to be designed like this, as well, since they don't have an ending in the usual sense, being sandboxes that expand as new content is added. Traditionally, the "distractions" all involve some kind of combat or another, but newer games in the genre have increasing focus on professions and gear tweaking. Long-runners even end up with multiple main narratives to choose from but as with offline games following this trope, they tend to be relatively weak.
Not to be confused with the Wiki Sandbox, or software sandboxes for untrusted code like browser-based Java games, or the OTHER kind of sandboxes. The one with sand. In a box.
The gamebook series The Fabled Lands (Fabled Quest in the US) was based around this idea, with hundreds of different side quests, some stretching across multiple books each of which represented a geographical area. If your character walked (or sailed) to edge of the map in one book he would end up in another, and there was no need to start the books in any order. Unfortunately, only six of the planned twelve books were published, leaving the world incomplete. Rumours of a Project Aon-style online revival with all twelve books remain unconfirmed. What has been confirmed however, is that the first 6 books are being re-released on the iphone sometime in 2010, and if interest is high enough, the other twelve will also be written and released.
Some of the Middle-Earth Quest gamebooks also managed this. Notably, A Spy in Isengard let you go anywhere on the map, and let you revisit the same location as many times as you liked (although, granted, there were a finite number of events that could happen at any location. There was a time limit, but only if you used the advanced rules, and even if you blew it, you didn't lose, you just got a less optimal ending; in the basic rules, Take Your Time was in full effect (that could actually be a minor problem, since it forced you to show up early for certain critical events). There was a goal, of course, but you could choose among multiple possible ways of accomplishing it. All in all, this was a very high degree of openness for a gamebook. Some of the other books in the series managed comparable levels of openness, but at least one, Treason at Helm's Deep, thoroughly averted this trope.
The first book in the series, Night of the NazgŻl, used the same game mechanics to achieve a similar Wide Open Sandbox feel, although with some wrinkles. As with Spy, there was a time limit in the advanced rules, although that again only determined the optimality of the ending. More peculiarly, many locations were functionally identical to other locations. For example, almost every map hex within the Barrow-Downs contained tombs that you could explore and loot. Each location text entry for the Barrow-Downs, however, referred you to one of maybe two or three encounter text passages, so if you thoroughly explored the entire Barrow-Downs, you would run into effectively the same monsters and the same loot, and the same text passages describing them, over and over again. So in effect, you were playing in a Wide Open Sandbox in which many places were completely identical to many other places.
This is waaaaaay Older Than the NES, having been a widely accepted style of play in the early days of Dungeons & Dragons. The "Wilderlands" series of third-party supplements by Judge's Guild, now considered classics, were particularly good at supporting this style of play. Many recent products that have come out of the "Old School Renaissance" revive this approach to the game, including at least two updated versions of the Wilderlands themselves. Most of these products are for older versions of the game or newer "retro-clones" thereof, but the "Kingmaker" adventure path for Pathfinder has elements of it too.
Stars Without Number is intended to support this playstyle. However, Word of God has it that since normal sandbox play is less than easy on the GM due to the necessity of designing potentially very many locations the players might never go, the rules were designed to make it easy to graft in modules from other games, especially second edition D&D, with a little reskinning.
The highly successful Source mod Garry's Mod is nothing but the sandbox. While there are game modes included, they came later.
Shadow of the Colossus subverts this sort of game: there's a giant, open world and nothing in it at first glance (and not much at second, actually).
Steambot Chronicles is a Japanese video game take on this, piloting a mecha, riding a bicycle, playing music or pool for money are just a few of the things you can do.
Betrayal at Krondor has an example of this type of gameplay done well - while you can literally explore nearly all of the available game world as early as the first chapter, the side material does not actually outweigh the central story, because you keep bumping into it wherever you go, and an incredibly strong central narrative is hailed as one of the game's greatest achievements. Also, the world map has the chapter number and current main objective on it.
No More Heroes subverts this by seemingly giving you a world to play in, but nothing to do there. You can't use any attacks in the over-world, you slide harmlessly off of people you run into, and your bike can't even cause the lousiest bit of collateral damage. It would seem like unintentionally poor design if the game's creators weren't so Genre Savvy. Not that there still is treasure to dig up and dumpsters to dive into for those who want to explore every nook and cranny of the game world.
The Civilization games avoid this (except when your starship is ready to make the trip from Earth to Planet) by declaring you will retire sometime in the 21st Century. After doing fine for several thousand years it's hard to see why you step down now. After raising a civilization out of the Stone Age, gardening leave is hardly going to cut it, is it?
The game sort of addresses this by saying in the end-message that it's your dynasty which is retiring, not one person who's been ruling right from the start.
Also, all the Civilization games offer an "extended play", which allows you to play indefinitely after a victory condition is met, albeit without any additional scoring.
Most Paradox Interactive games are like this. You start out in a more-or-less historical situation (say, the world in January 1936), then let you make your own decisions and set your own goals. Each has a "victory" condition, but these are often ignored by the fanbase. Playing as an especially hard country (say Serbia), it may feel like victory just to finish the game intact.
The unexpectedly goodThe Incredible Hulk: Ultimate Destruction is exactly what it sounds like. You're given a few different locations to pretty much smash up as you see fit when you don't feel like progressing the story. Among other things, depending on what your whim is at the moment, you could surf around on flattened buses, play a home run derby with people as the baseballs, or just have fun running and leaping around a city or desert, enjoying how fun it is to control the Hulk.
In RollerCoaster Tycoon II, after careful selection of objectives ("build 10 fancy coasters") and money ("none needed"), you can pretty much ignore the game and spend hours designing the perfect shrubbery.
RCT 3 finally adds an actual Sandbox mode with no constraints but the boundaries of the park, the game's physics limitations (which can mostly be circumvented with cheats), and your own creativity. Well ... that, and the size of your computer's memory/processing power (it's quite possible to overwhelm average machines with too complex a park after building too much).
The entire trade-simulation genre (notably the Patrician and Port Royale series) is all about this trope. The only requirement is "don't go broke".
Sid Meierís Pirates! (and its remake... and the remake of the remake), sorta. They're extremely open ended, almost to a fault, but they also have what amounts to a time limit in that your pirate ages over time, and eventually has to retire. Many critics believe this aging feature is the game's biggest flaw.
The games based on the 2007 Transformers movie have Wide Open Sandbox worlds for the player to roam in-between missions, which are incidentally optional, but recommended to unlock content. It's somewhat like Grand Theft Auto... except that you happen to be the vehicle.
Genesis version of Shadowrun is an example. There is an overarching plot stuck somewhere in there, but messing about in the gameworld is too much fun to care.
The only mandatory levels in the second half of Final Fantasy VI are Figaro Cave/Castle, Darill's Tomb, and Kefka's Tower. Everything else — and there's a ton of stuff — is an optional sidequest.
The surprisingly goodSpider-Man 2 movie tie-in game gives you free rein in Manhattan after the tutorial. You can go anywhere, do anything, and generally do whatever a spider can, enjoying how fun it is to control Spider-Man in the process. Want to set the record time for crossing the island end to end? Sure! Want to perform some epic-level Le Parkour? It's almost required! Want to swan dive from the Empire State Building or Statue of Liberty, only to save yourself at the last moment? (Or not save yourself at all?) As many times as you want! Oh, sure, you have a little white marker pointing you toward the plot, but you can go there any time. Besides, to finish chapters, you need Hero Points, which are earned doing just about everything you do wandering about the city: beating up thugs, solving crimes (which involve beating up thugs half the time), locating Hint Markers to contact the snarky narrator (Bruce Campbell), completing speed challenges, and... delivering pizzas.
Likewise with Ultimate Spider-Man (which is based off the comic book series of the same name) which includes an original story and a more comic book feel of its predecessor.
Most of the Jak and Daxter games permit you to keep wandering around as long as you like, completing the little optional missions, beating up enemies, or just picking up the "Unlimited Ammo" cheat and blowing up everything everywhere.
Jaws Unleashed for the PS2 and Xbox was set in the wide open ocean, based off of the Jaws film series. You played as the shark. Unfortunately, the idiotic controls made it a lot less satisfying than it should've been. Still, you get to fight an Orca Whale. In the middle of an ocean park. And tear it in half. And there's a fight with a giant squid inside of an oil rig. And the swimmers are all Made of Plasticine.
The Sims. The game manual itself mocks the reader for asking how to win! You can't win. There is no "win," just endless torment of your immortal pixel people. In the later installments of the series, however, the introduction of aging and Lifetime Wants allowed the players to create goals for themselves and their Sims. You still can't actually win though. Just play forever.
In FAMOUS for the PS3 is a superhero game that casts the player as a bicycle courier who develops electrical powers after being caught in a bomb blast. There are mandatory Story Missions as well as optional missions, but you can take as long as you want to work your way through the story. Your actions determine how the civilians view you and what new powers you can unlock. It's like the mutant offspring of a three-way between Grand Theft Auto, Assassin's Creed, and Fable.
Infinity: The Quest for Earth fits here like a glove. It will have over 200 billion star systems, and everything (stars, planets, moons, asteroids...) is life-sized.
The Original MechWarrior PC game had a storyline about restoring your family to a throne. But the rest of the game was so open ended many players never even knew there was a storyline in the first place. Mechwarrior 2: Mercenaries and Mechwarrior 4: Mercenaries allow the player to take missions as they please, such as entering themselves in the Solaris Arena matches in Mechwarrior 4 for extra money and fame with one of the two factions in the civil war. Once your fame is high enough with one side, you'll unlock their plot missions and ultimately the endgame mission.
Red Faction: Guerrilla takes place on Mars: a literal and figurative sandbox. There are numerous side missions to help in the liberation of Mars. Interestingly the game will actively start optional side missions when you are out driving through the badlands and settlements, as well as allowing the player to choose specific missions. Completing side missions provides resources which help in the unlockable core story missions. The story missions are generally more elaborate and spectacular.
The Legend of Zelda I is considered to be a very early interpretation of the genre, being that you can access (almost) any dungeon from the start (on the proviso that you can find it) whether you have the new weapon/Plot Coupon from the prior dungeon or not- later titles are much more restrictive and linear by comparison, despite still allowing you to bomb around the overworld more or less at will.
At E3 2014 Eiji Aonuma revealed a Zelda game for Wii U and announced his intention to replicate the openness of the original game now that Wii U will allow them to do that in 3D.
The licensed video game of Toy Story 3 features a Toy Box mode, which basically lets you mess around with the characters however you want. It's intended to simulate playing with toys in Real Life.
EA's Skate series gives you a city and a skateboard. Especially improved after the first, with the addition of on-foot controls to Skate 2 and improvement of said controls in Skate 3. You can get some amazing footage for Skate.Reel just messing around. Not to mention the Film/Team Film and Hall Of Meat challenges, which you can do by skating up to any location and just pulling off the requirements. It becomes a challenge of seeing where and how the challenges can be done.
Midnight Club Street Racing and its sequel had a "Joyride" mode where you could simply drive all over town, exploring, with no time limit or objectives - even with a friend in split screen!
The Animal Crossing series has TONS of stuff for players to do, such as paying off the price for the house to expand it, collecting all fossils/bugs/fish/paintings, collecting every single item or piece of fruit as possible, etc. Yet, the player can choose to not do any of those things and just spend time wandering around town or go online and visit a friend's town.
The Falling Sand Game, in which sand falls and you... draw things for it to fall on or into. Among other things. Also a literal example. It being a Wide Open Sandbox is mocked in the song ("Don't know how to win at all...")
Mabinogi, a free Korean MMORPG is far more of a sandbox type game than is typical for the genre. There is an overarching story, and several major side stories; but the majority of the game's content can be accessed outside of the storyline.
The ARMA series sets its missions on "islands," which are huge yet fully rendered terrains (in the style of games like Grand Theft Auto or Assassin's Creed) with no separate loading times, but in turn the editor allows for an incredible variety of what the player is actually to do within these 'islands', and there are several player-created missions that are essentially Grand Theft Auto: The MMORPG.
The mainland in Ryzom is so exceedingly huge that some new players, upon completing the tutorial, will delete their character, make a new one, and stay on the tutorial continent for a while longer this time.
Spectrum Holobyte's Vette may have been the very first open sandbox driving game, and was one of the first true 3D driving games. The Tour mode allowed you to freely roam the city and quick-travel between landmarks.
The tie in games for Harry Potter and The Chamber of Secrets, Harry Potter and the Prisoner of Azkaban, Harry Potter and the Order of the Pheonix, and Harry Potter and the Half-Blood Prince.
Kenshi is similar to the above example - there's nothing but the sandbox, and it's up to you to thrive in it or die trying.
Dragon Age: Inquisition aims to combine the traditional narrative focus of the Dragon Age series with an open world design, by tying the story progress to exploration and particularly to the growing power of the title organization. Story decisions and combat still form the gameplay core, however, so the sandbox aspect is minimized.
Parodied in an episode of South Park where the boys loafed around playing World of Warcraft and a majority of the episode was even made up of actual gameplay. After they spent most of the episode working towards stopping a "griefer" who kept slaughtering other people's characters, the episode ended like this:
Cartman: We did it you guys! We're totally heroes! Kyle: That was such uber pwnage. Stan: I can't believe it's all over. What do we do now? Cartman: What do you mean? Now we can finally play the game. Kyle: Oh yeah.
The King of the Hill episode "Grand Theft Arlen" has a couple of community college game programmers make a GTA clone called Pro-Pain!, starring a copy of Hank mainly because they thought he was funny. Initially Hank is disgusted by the violent content ("Where's the button to turn myself in?"), but he starts getting into it when he discovers that the sandbox element lets you do good deeds like stopping purse-snatchers and robbers.