Monty Hall (note the lack of a "U") was the host on the game show Let's Make a Deal, which was cancelled before some of you were born. The show could give away massive amounts of prizes to the lucky (or cunning or destined-to-win or however they pick winners on game shows).
A Monty Haul campaign (with a "U") was the generic label for a Game Master (and his/her campaign) who would run adventures that were like game show giveaways, except the questions weren't as hard. Players would end up staggering under the loads of gold and gems (except the encumbrance rules often were ignored as well) and cherry-picking which magic items they wanted to keep because they had so many to choose from. Think of Conan the Barbarian with a Star Destroyer.
Also, in the first and second editions of Dungeons & Dragons, you got experience based on how much money you looted, one to one. So the Monty Haul characters would also end up with stratospheric levels, which led to situations like characters assassinating gods like Thor to gain their nifty weapons.
This is a situation greatly deplored by most gamers, and discouraged sternly in the gamemaster's section of all later games, but it is assumed that everyone went through this stage at some early point in his gaming "career."
In some other cases though, the Monty Haul game master may be a case of Suspicious Game Master Generosity since sometimes they may take it to bring upon an already dangerous foe and give him a slew of new tricks or has their stats tweaked for whatever to get ready for a tougher battle.
Naturally, no troper here ever descended so low.
Many video games seem to either encourage or require the sort of player (or the sort of game play) who would take a broom to a dungeon, to make sure he swept up all the coins.
The opposite to this is the Killer Game Master, who delivers death and horror to the PCs in place of treasure and godlike power.
This doesn't have much to do with the Monty Hall Problem, a probability puzzle whose name also comes from the game show host.
Examples:
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Comics
Knights Of The Dinner Table the seasoned regular Game Masters generally either run balanced campaigns or go the opposite direction of this trope. However, when Bob or Dave step behind the screen to run a game, it goes this route.
Literature
This trope plays an unusually huge role in the book, Slathbog's Gold.
Each edition is its own game, balanced around its own set of expectations.
This trope cannot be described in absolute terms, but is entirely relative to the expectations of the game.
While each advancing edition has objectively made magical items more and more common, they have also become increasingly required in order to function at the expected level. Any magical items at all were only a bonus in AD&D, while fourth edition made a steady treadmill of ever-more-powerful weapons and armors a mandatory part of not dying horribly. Trying to apply a fourth edition approach to AD&D would indeed result in Monty Haul gameplay, but someone taking the AD&D approach to fourth edition would instead be a Killer Game Master.
The LegendSystem's magic item rules are specifically designed to prevent this - the progression is the same regardless of the DM's inclinations.
Webcomics
Referenced briefly but used incorrectly in Goblins. Forgoth mentions that Minmax cannot simply walk into a blacksmith shop and buy a magic sword, as that would be too Monty Haul, but that's exactly what PCs are expected to do with their treasure. (If you look in the DM's guide, it clearly says a PC can generally buy any magic item on the list, though they need to look for a seller.) Monty Haul only comes in when PCs have too much money for their level.
However, at the same time, the campaign itself definitely is Monty Haul. All the heroes are low-level, but have improbably powerful magic items that even the playing field. The paladin has an axe that acts as the can for a world-destroying demon, the cleric has a spear that multiplies when thrown and then returns to his hand, and the minmaxer recently got a sword made out of oblivion. Which is perhaps fair enough, considering everything else.
Jerry Holkins (Tycho) accuses Mike Krahulik (Gabe) of this in Penny Arcade. Tycho decides to balance it.
Western Animation
Dee Dee from Dexters Laboratory, after taking over DM-ship of Dexter's game. Considering that Dexter was a Killer GM, they were much happier with Dee Dee.
Video Games
The Diablo series and the majority of its clones tend to be like this in the end game. Bosses and major loot caches will often release a screen-filling fountain of gold and enchanted gear- from which players will pick the one or two very best pieces and leave the rest lying on the floor. At early levels, however, the player will want to keep anything that's better than the standard vendor gear. For a game where the whole point is to constantly upgrade your equipment, the progression is fairly even.
High Rate Private Servers of any given MMORPG are more often then not Monty Hauls meant to fuel a player's armory with the weapon needed to fight big bosses and most of all each other.
A lot of user-created modules or modifications for The Elder Scrolls and Neverwinter Nights are simple dispensers of XP and loot. Many of them who have actual plot can also fall into this due to being made by non-professionals.
Many badly made designs for Unlimited Adventures shower the characters with money. One design in particular (From Beggars to Heroes) was content to throw thousands of money at you for the meekest reason. (For example, as poor beggars in the starting town, you can walk around and meet an unlimited number of rich people every few steps who will shower you with riches every time.) Oh, and you get experience for these, too.
S.T.A.L.K.E.R. also had its share of mods which purported to 'improve' the gameplay but in the process dumped some of the game's best weapons and gear into the player's arms near the very beginning. Kind of ruins the 'scavenge and survive' experience when your stuff is already better than everyone else's...
Dungeon Siege encouraged this by adding pack mules and Traggs, creatures with above normal carrying capacity, but minimal fighting capabilty. Naturally it's a trade off between combat power and hauling size.
Fable, while in itself the game isn't a Monty Haul, if you spend an hour or two doing a job to afford some real estate, the returns on the investment are used to buy more real estate (which pays you every five minutes) eventually become so huge that you have so much money you have no idea what to do with it all.
Fallout 3's Operation Anchorage DLC takes the player through a short, relentlessly linear, and relatively easy (depending on character build) campaign, at the end of which you're awarded with a treasure trove of goodies not available anywhere in the core game.
Ditto Mothership Zeta, and the steel ingot fetch quest in The Pitt.
Many Fallout players consider Fallout 3 to be a Monty Haul campaign in general, especially compared to previous installments and Fallout New Vegas.
The Black Jack tables in New Vegas (and more so the slot machines if you have high luck and too much time on your hands) are ridiculously easy ways to clean out every casino that has these minigames and earn up to 100,000 caps.
The treasure at the end of the Dead Money DLC is 37f gold bars worth 10,000 caps each (though each weight 35 lbs and there's a time limit before your collar self-destructs). However, that is a subversion: not only are you expected to leave the vast majority of them there, they are also worth less (per pound) than many other items. The weapons and equipment found as well are good, though not completely game-breaking. Finally, by beating the DLC the player gets access to a Matter Replicator vending machine that takes the special Sierra Madre chips which you get a bunch of every 3 days—most notably they give as many Weapon Repair Kits as you can pay for, which you normally have to get by the rather exhausting process of searching around for specific crafting parts.
Dragon Age: Origins has the "Soldier's Peak DLC". Accept a quest, spend 10 minutes mowing down underpowered Mooks, proceed to a vendor offering the best weapons and armor in the game. Granted, you might have to spend another 10 minutes mowing down money spiders and selling off their Vendor Trash... Still, the DLC takes about an hour to get through and is fairly difficult. And by the time you'll be able to afford and use that vendor's high-level equipment, you'll damn well need it.
Dragon Age: Origins – Awakening also throws more money at the player then they may know what to do with. This is in stark contrast to the original campaign, where money is quite tight (barring DLCs and a single developer oversight that requires a lot of loading screens). Players new to the expansion may be shocked to find quest related NPCs requesting of them things on the order of dozens of gold pieces to commission some sort of order—a small fortune in the original campaign, pocket change in Awakening. With so much money available, you wonder why the Warden doesn't just hire a couple hundred mercenary soldiers to build up Amaranthine's otherwise vastly outnumbered forces.
The Architect Entertainment buildings in City of Heroes were created to allow players to write their own story arcs for other players to enjoy. Among most of the community, however, it's better known for its "farm" missions, as they're called. It's gotten to the point where some farms have earned "Hall of Fame" status for having so many favorable ratings among players. The earliest farms would take advantage of exploits to allow characters to go from creation to level cap in a single day of beating up defenseless enemies for disproportionately high XP. Naturally, the devs did not take kindly to this, and closed such loopholes whenever they find them, even banning the most egregious exploiters of them. Newer farms aren't quite as efficient, and are not cracked down upon as forcefully, however.
In the MMO Star Trek Online, it is ridiculously easy to gain the best gear and equipment for little risk via earning special ingame currency, and via crafting said gear using easiy farmed resources called "data samples". Also any dropped gear that's rare but you don't need can be sold on the player exchange for insane amounts of ingame currency. Many players become multi-millionaires this way, making said currency practically worthless - save for some special items like rare bridge officers that boost your in-space abilities (only one type which is so rare that it sells for millions on its own).
Subverted now for crafting when the game went free to play, as crafting high level items now requires Dilithium. Which can only be produced in quantities of 8,000 a day, and most items require around 12,000 Dilithium to craft. Only players willing to spend real money on the game to trade it to others for Dilithium can really craft the high level items.
Firkraag's Dungeon (Chapter 2) in particular had a metric ton of unique loot, including a +3 bow, a Dragon helm/shield/sword/scales (all with bonus resistances), Full Plate Mail +1, the best two-handed sword in the game (Carsomyr), a shield cloak, not to mention a ton of ammunition and various minor weapons. Any party that ventures in will find enough gear to make a killing on the resale market.
For a savvy party, going into Watcher's Keep in Amn and stealthing your way past most of the enemies on the first two floors will net you tons of +3 and +4 weapons and ammo to sell, long before you'd get anything else in the main plotline that compares.
