In Digimon World 4, each character can set up to three equipped weapons at a time. If you want to use that shiny new sword you picked up from a defeated enemy, you'll have to go back to base and equip it at the Digi Lab. Justified in that this is the Digital World and the Digimon probably have to load the weapon data.
BioShock handles this a little oddly. In the firsttwo games the player has a Hyperspace Arsenal as far as weapons go, but the player is limited to only a select number of plasmid powers at a time, with additional slots costing precious ADAM. BioShock Infinite, however, decided to give Booker acess to every Vigor at once while guns were limited to just two at a time.
Brink limits the player to a primary weapon, a backup weapon, some grenades, a few class-specific weapons (such as specialist grenades for a soldier and landmines for an Engineer), and possibly a knife if they use a pistol as one of their weapons.
The Call of Duty franchise follows this, limiting the player to any two weapons, with a third slot for a pistol in the original game, and some non-gun odds and ends (generally frag grenades, adding smoke grenades and other alternatives starting from the second game; the maximum is two grenade types and two other thrown/placed explosives).
Fire Warrior has the player able to use two weapons and the bonding knife, but one of those weapons (the standard-issue Pulse Gun) cannot be replaced, forcing the player to drop any good gun they find to get another one.
Left 4 Dead: You can carry one primary weapon (generally rifles and shotguns) from a selection, one secondary weapon (either an unlimited ammo pistol or two, a single bigger pistol, or a melee weapon) and one equipment item of each type (only one thrown explosive such as a pipe bomb or a molotov, choosing between a medkit and a defibrillator, etc.).
Metro 2033: You can carry a primary weapon (one of a selection of assault rifles or an SMG), secondary (shotguns/pneumo guns), a revolver, an Emergency Weapon knife, a medkit containing five syrettes, five pipe-bombs of each kind and five throwing knives.
Perfect Dark Zero has a variant incorporating some Grid Inventory: at the start of each mission, Joanna is alloted four slots for weapons. Each weapon available for her to start with (you have to finish a previous mission holding them for them to become available) takes up one (small ones like pistols), two (submachine guns), or three (e.g., rifles) slots.
Rise of the Triad: The player can hold at one time any number of the infinite-ammo bullet weapons (of which there are three: single pistol, dual pistols, and machine gun) and only one of the limited-ammo missile/magic weapons.
Soldier of Fortune 1 and Payback only allowed the player to carry a max of three guns(depending on their size) and a knife, but the second game had a significantly higher loadout limit, e.g. Mullins may lug around a sidearm, shotgun, assault rifle w/grenade launcher, light machine gun, and sniper rifle at once.
Crysis uses a weird version. The player has a pistol slot that can go Guns Akimbo (there's only one kind of pistol in the first game, though Warhead adds an SMG that can be dual-wielded), an explosives slot for remote-detonated charges and a missile launcher, and a long arms slot that can hold assault rifles, submachine guns, sniper rifles, and shotguns (pick two).
Far Cry 2 allowed the player to carry a machete, one primary weapon (Typically a rifle, though machineguns and submachine guns are available), one secondary weapon (Typically a pistol, though machine pistols and a grenade launcher are available) and one special weapon (Heavy weapons).
Far Cry 3 allows the player to carry up to four guns and a knife, as well as grenades and explosives, though the player has to unlock the ability to carry even a second gun by hunting and skinning animals.
Years beforeHalo implemented this trope, Trespasser gave the player the ability to carry only two items: one in hand, and one stowed.
Borderlands plays with this somewhat. You can have up to 4 guns and no more set to your hot keys. However, there is limited space in your backpack to carry extra weapons, and the ones in there are probably just Vendor Trash. The limited space disappears as you progress, though, since you can upgrade your backpack to hold more items - 3 per upgrade. The extra guns in your pack aren't feasibly usable as extra firepower, however, since you're still limited to 6 ammo types (7 in the first), and the game doesn't pause in multiplayer while you swap weapons. A rare and well-executed combination of Hyperspace Arsenal and Limited Loadout.
In Splinter Cell Conviction, Sam and the two co-op characters Archer and Kestrel are only able to carry one main weapon (a rifle or shotgun) and a pistol. Sam can carry every type of gadget at once but the co-op characters get just any two at a time.
Alone in the Dark 2008: Carnby's inventory consists of what he can hold in his hands and the very limiting confines of his jacket, although you can also keep caches of things scattered about whatever stage you're playing in.
Dead Space: Isaac can only equip 4 weapons at a time. The rest must be either kept within the safe (accessible from any of the Ishimura's store consoles) or sold to those stores to free up space.
In the Commodore 64 game Project Firestart, you can carry two laser rifles at a time, and there's no way to recharge or swap ammo. If you want to pick up a brand new rifle from the armory, you need to empty out one of the ones you're carrying.
In Transformers Prelude To Energon, you could equip up to four Mini-cons at a time when going into a level, but only two of which (assigned to the R1 and R2 slots) were actual weapons. When you collect a new minicon you have the option to swap it in, but otherwise you have to exit the level and re-enter if you want to change your current set.
The first three entries in the Gears of War series give the player access to two primary weapons, a pistol, and grenades on a cross-shaped menu. The fourth game, Judgment, ditches this menu, making pistols just another weapon you carry in your two slots, and grenades throwable at any time with a press of a button. It also sped up weapon switch times and melee speed considerably, and its multiplayer ditched the firepower boost from a perfect active reload, forcing players to utilise their limited arsenal to its fullest effect rather than simply relying on theGnasherShotgun to carry them through battles.
Warhammer 40,000: Space Marine has two "special" weapon slots (that can't be used when wielding a thunder hammer or wearing a jump pack) in addition to the chainsword + bolt pistol and the bolter. Special weapons include a storm bolter, plasma gun, remote grenade launcher, sniper, bigger sniper, meltagun... You can also replace your melee weapon from the standard Astartes pattern chainsword to anything from a combat knife to a power hammer.
The first three Syphon Filter games had hyperspace arsenals, but in the second trilogy, they switched to a limit of one of each weapon type(melee, grenade, shoulder arm, sidearm, and auxiliary arm).
Max Payne 3 limits the player to carrying one rifle-grip gun and two pistols.
XCOM: Enemy Unknown: Every one of your troops gets a primary weapon (the type of which is determined by their class), a secondary weapon (a pistol, unless they're a Heavy in which case they get a rocket launcher instead) and a slot for extra utility items like medkits, grenades, or armor. Support troopers unlock an extra utility slot once they reach a certain rank.
The Enemy Within DLC has the MEC Troopers who have a primary ranged weapon and a secondary, limited use weapon that can be either medium-range (flamethrower) or close-range (Kinetic Strike).
Done various ways in Mass Effect, which also makes a point of averting Informed Equipment as your weapons appear in a collapsed form on your character model.
Mass Effect 1, being more of an RPG that happened to have guns in it than a shooter, had a standard RPG Hyperspace Arsenal inventory (hard-capped at 150 items.) However characters were limited to one of each weapon class equipped at a time, were usually only proficient with a couple weapon classes, and couldn't change which weapon went where during combat.
Mass Effect 2 only allowed most characters to carry one copy of two weapon classes. Shepard got a heavy weapon as well and depending on his/her class might have additional weapons, and during a particular plot mission could either gain an additional slot or get an infinity plus one gun version of his/her primary weapon (for example, an Infiltrator could add the M-98 Widow Anti-Materiel Rifle to his/her sniper rifle selection). Soldier classes could choose any of the three infinity plus one guns, due to being able to carry all weapons except sub-machine guns from the start, and the game engine only being able to handle Shepard knowing how to hold up to five guns at a time.
Mass Effect 3 restricted Shepard's party members to two weapon classes out of five, (and are treated as always being 400 points overencumbered) but, as seen in the page picture, Shepard had access to all five slots. Using all five weapons made his/her powers recharge extremely slowly but nobody actively stopped you from doing it. Soldier Shepard can level up to carry a hell of a lot to compensate - Infiltrator and Vanguard Shepard can also carry more than Shepard's other non-Soldier classes/the multiplayer crew. Also, Shepard's class-specific perks only affect certain weapon types. Heavy weapons change to Throw Away Guns that are placed on the map, replace the equipped weapon when picked up, then tossed when they run out of ammo. Multiplayer characters also had access to all 5 slots, but are limited to a choice of two weaponsplus a weightless Cobra rocket launcher (classed as a missile launcher despite not locking on) that they always carry, even when empty, and are also under the same weight limitations as Shepard, with carry capacity roughly equal to that of his/her Engineer/Sentinel/Adept classes.
ME3 is an interesting case, in that the game expects you to be over-encumbered at any given time, and avertsCritical Encumbrance Failure while doing so. The base cooldown for any given power is actually the cooldown time you get if you're exactly 200 weight points over your base capacity (including passives), and that time is added to or deducted from by up to 200% either way, with the minimum times being gained from not being over-encumbered at all, and the worst times coming from being a whopping 400 points or more over-encumbered. Justified by the use of element zero-powered mass effect fields compensating for a fighter's inability to carry their load without slowing down or harming their muscle or skeletal structures, at the cost of taxing their tech generators/biotic abilities.
In The Witcher, you can only carry up to four weapons at a time: two Witcher swords (steel and silver), a short weapon (such as a dagger or a small axe), and a heavy weapon (e.g. a large axe or a morningstar). You cannot carry any weapons in your inventory at all.
Automated Simulations' (later Epyx) Temple Of Apshai. Your character could only carry one weapon (a sword) at a time. If you wanted to pick up a sword you found in the Temple, you had to drop (and lose) the one you were holding.
Wide Open Sandbox
In Dead Rising 2, you are limited to carrying only a certain amount of items in your inventory, as well as certain items you can't keep in your inventory while equipping another item (like the chainsaw). Fortunately, as your level increases, so does the amount of items you could carry.
A new mechanic introduced in Grand Theft Auto: Vice City and carried on from there is weapon categories, from melee weapons to shotguns to thrown weapons to gifts. It's more than one would expect a normal human being to be able to carry, but it does prevent situations like carrying both a rocket launcher and a flamethrower as was possible in Grand Theft Auto III.