The BattleTechHumongous Mecha wargame has rules to allow the 'Mechs to pick up arms and limbs that have been shot off other 'Mecha (although, there's nothing stopping a 'Mech using its own amputated leg, come to that).
If your 'Mech has hands and is in a forest hex — hey, insta-club! Sure, it's only good for one use, but they're like Doritos: crunch all you want, they'll make more.
The Orks of Warhammer 40,000 have their entire civilisation based around this (and getting More Dakka). Everything is improvised, and everything is a weapon.
An Imperial assassin of the Eversor temple was once assigned to eliminate a planetary governor and his family after it was discovered that the colonial leader was financing pirate raids against his neighbors. Attacking during a family festival, the assassin ended up butchering all three hundred and seventeen members of the governor's clan, ages ranging from one hundred and forty-two to three weeks old, though he had to resort to using a silver soup spoon and a butter knife before he was finished.
The annihilator pattern of the Predator tank originated as this, ironically. It was the result of a long ranged long fang assault against a group of chaos space marines, where they were unable to get in range with their lascannons, and the predator destructor they had at hand were not an effective enough tank-hunting weapon. So what do they do? Rip out the weapons of the tank and link their lascannons there instead! While the Adeptus Mechanicus were enraged at even thinking of modifying their tanks, the pattern soon grew in popularity, and soon became one of the most popular predator patterns in the 41st millenium once other chapters started copying this idea.
The Apocalypse datasheet for the Land Raider Ares has a backstory a bit like this. The Dark Angels (I think) were bogged down in siege warfare (which doesn't answer why they didn't just have their strike cruiser torpedo the enemy fortress into scrap), and they had the techpriests build them a weapon designed to penetrate the rubble-strewn defenses. A day after the Areses were deployed, with their heavy flamethrowers, epic machine guns, and massive short-ranged artillery weapon, the enemy surrendered.
In the spin-off game Necromunda, there's a grappling hook, which "Is not really a weapon, but it's inevitable that someone will want to use it as one." It's the same strength as a normal gun.
In the Urban Arcana setting for D20 Modern, there is an advanced class called the Street Warrior that gains proficiency with improvised weapons and at later levels can do extra damage as if it were a size category larger.
In fact, most roleplaying games allow characters to wield anything they can reasonably lift as weaponry. Generally, however, unless the item was designed to be a weapon, the attack is penalized.
In In Nomine, Malakites of Creation (and any other Angel with their Choir Attunement) explicitly have the ability to pick up anything and use it as a weapon, the effectiveness determined by their stats rather than the object in question, the explanation being "they fight creatively." Naturally, this leads to plenty of Once Killed a Man with a Noodle Implement.
In Scion, a character with sufficient Epic Strength can use anything ranging from a motorcycle to a nuclear aircraft carrier as such.
Scion's pseudo-progenitor Exalted has a lot of the same, only in one case, it's so much cheesier. In 2nd ed, the core rulebook's set of Exalted, the Solars, have their titular martial arts style, Solar Hero Style, have a minor focus on improvised weapons. The Lunar Exalted are much, much worse. Not in Lunar Hero Style, but because anything they can pick up, they can make a perfect attack with using one of their Charms (a perfect attack just hits, unless you use a perfect defense to stop it). Including doing so with improvised versions of weapons they have absolutely no idea how to actually use in the first place. For Full Moon Lunars under their anima effect, this includes picking up decent-sized buildings and hurling them so very hard that the attack is unblockable and undodgable. Or as a melee weapon. Or, if you have a ballista on hand, as a ballista bolt, even if you don't know the first thing about siege artillery.
Meanwhile, a good chunk of the Exalted types get Melee and Thrown Charms. Under the right circumstances, these can apply to dessert forks and fruit. Similarly, the Sidereal Exalted get an Archery Charm that allows them to turn anything shorter than their arm — such as a table leg, a handful of sand, or a shout — into an arrow, granting that arrow properties of the material you used (you aim for someone's face with a sand arrow, they're gonna be blinded).
Besides the normal improvised weapon rules GURPS: Powers spends two whole pages on the effects of using improvised weapons that are bigger than you are (like cars, or I-beams).
And in the Dungeon Fantasy setting Ninjas can temporarily make improvised weapons just as effective as proper weapons.
Always at the cutting edge of tavern brawl technology, 7th Sea features a school of fencing dedicated to the use of improvised weapons. Note that it doesn't give any advantage to combat with real weapons. Rather, at a low rank it allows the fencer to wield anything as if it were a weapon — anything at all. More amusingly, at higher ranks the character can become more dangerouswith a broken bottle than he would be with a sword.
In Paranoia, Bouncy Bubble Beverage cans explode if you shake them enough, making them popular among citizens not cleared to use actual grenades.
One Dungeons & Dragons prestige class in 3.5 edition, the Drunken Master, has the special ability to use any improvised weapon, but deal damage as if he were attacking with a Monk's unarmed strike (At the Monk level when this ability is possible, you may be dealing 1d10 damage with fists; the improvised weapon damage die is 1d4). This ability has one caveat; It is only active when the character is drunk. A secondary ability of the Drunken Master is to be able to drink a pint as a move action.
The Complete Warrior splatbook sort-of creates the ability to treat specific improvised weapons as Exotic Weapons (one of the Prestige Classes includes Exotic Weapon Proficiency - Manacles as a bonus Feat for a Class Feature). The back of the book also includes rules for improvised weapons such as table legs and barrels, giving rules for transmuting weight of the object into damage dice, and specifying that sharp objects deal double damage. Exotic Weapon Proficiency - Boat Anchor has never looked quite so fearsome.
In 4th edition, the Arena Fighter can use anything as an improvised weapon with a fair degree of skill. There's also the Belt of the Brawler, which allows the wearer to use any improvised weapon as though it were a club. Combine that with the Iron Soul Monk and you're basically Jackie Chan in a ladder factory.
Pathfinder takes improvised weaponry to a whole new level with decent damage dealt with improvised weapons in general (based on their size compared to actual weapons), rather easily accessed feats to remove attack penalies when using improvised weapons, and a lot of classes have archetypes that give some of these feats for free with other benefits. One notable example is the Makeshift Scrapper, a modified Rogue who forgoes trap detection to focus on fighting with anything available, even getting bonuses on attack rolls with improvised weapons rather than penalties.
In Feng Shui, the Everyman Hero character type gets a bonus when fighting with any improvised weapon.
Mutants & Masterminds includes rules for improvised weapons, as well as a couple of feats to make them more effective. However, a super-strong character is likely to destroy whatever they were holding after a swing or two.
The weapon charts in Post-Apocalyptic Hero (5th edition Hero System) include such items as parking meter maces and "swords" made by sharpening the edge of a traffic sign pole. Well, it is post-apoc.
Changeling: The Lost has the Bloodbrute Kith, who gain the ability to fashion anything they find into a crude weapon. The book mentions an Ogre ripping a stop sign out of the ground and fashioning it into a great axe as one example.
Golems in Promethean: The Created can gain the ability to double or triple their (usually already formidable) Strength for the purposes of lifting — taking them into the realm of picking up I-beams, trucks, or 16-ton weights. An upgrade to that ability lets them swing or even throw these massive objects without penalty, with predictable results.
Starblazer Adventures, based on the 1970's-80's British science fiction comic book.
The "Whatever's On Hand" stunt allows the use of the Fists skill when fighting with improvised weapons.
The "Anything Goes" stunt allows the use of any kind of improvised weapon without penalties.
In the Palladium game Ninjas and Superspies, there is a martial art focused around the ability to turn anything you can pick up into an effective improvised weapon.
Warhammer has this in both ranged and melee versions. In melee, Giants tend to wield whatever's handy, such as a tree with a few swords driven through it. At range, the Ogres have a fine line in stuffing scrap into Leadbelcher cannons, and the first war machine they had, the Gnoblar Scraplauncher, consists of a collection of weapons too small for Ogres but too large for Gnoblars, dumped into a trebuchet arrangement and fired. The Scraplauncher goes so far as to have the Killing Blow special rule, to represent the chance of one of the swords landing point first.
In the board game The Awful Green Things From Outer Space, when battling the eponymous Things, most of the potential weapons available to the crew of the spaceship Znutar fall into this category; the only things that were originally designed to be weapons are Stun Pistols and Gas Grenades.
Inverted with the Cargo Launch ship in Star Realms. It was originally developed as a combat drone, the Star Empire used it for hauling cargo instead. Which is a surprise - as the Empire faction tends to focus mostly on combat.
Every native Iote weapon and device in Rocket Age is an improvised weapon, made from whatever scrap they were able to find. Don't laugh though, their zip guns, crossbows and slingshots will end you if you underestimate them. Io might have the lowest technology level of the entire solar system, but the Iotes have more natural technical aptitude than practically anyone else.