If the game feels really generous you might even get a few basic Standard RPG Items in your inventory.
Weapons will be the default type for the character's class.
If there is more than one type of armour slot (outside of just "armour" for example for helms, gloves, accessories) then expect the more advanced ones to simply be empty. If you start off without a weapon, this may well be the only point in the game where you don't have one equipped.
Sometimes the character will start off completely unarmed. In this case the plot (or a handy chest) will provide the Starter Equipment. Probably during a tutorial.
Characters that join later in the game don't count since they always have generic equipment for that level (or even better equipment if you're lucky). Sometimes, at the beginning of the game, you'll be given something better to use before this trope catches up with you. Keep in mind that items that aren't upgraded or replaced as the game goes on and just serve to introduce game mechanics are not this trope, since they only act as an in universe explanation for that ability (so healing items which you'll be able to buy better versions of count, as do equipable items, but a PDA you're given that serves as your pause menu or a pair of gloves that enable Video Game Stealing but are never replaced or upgraded does not).
Compare Emergency Weapon for an early piece of equipment that has some slight advantage (typically being unbreakable or never running out of ammo) to make it a backup rather than just a springboard to later weapons and the equally useless Vendor Trash. If the hero starts out on an important quest but still gets no help with getting better equipment then this is With This Herring. Using it may be the aim of a Self-Imposed Challenge. This can overlap with Weapon Jr..
In most Real-Time Strategy games with a campaign mode, the first mission features one or two basic types of troops and may not even feature your Worker Unit.
In Ragnarok Online you start off with a shirt as your armor and a basic knife, both of which are extremely ineffective. However, doing the tutorials gives you a full set of much better equipment that will easily last you to the point where you can change your job.
In World of Warcraft all classes, with the exception of Death Knights (since they start the game at a higher level), start with some basic low quality items and in many cases a cosmetic shirt. Said shirts used to only be available through character creation (or by buying or trading from another player) until a vendor selling them was added in the game in the first expansion. Death Knight gear is high quality and unique, although there is some higher level gear with the same models.
It's possible to obtain copies of the Blood Elf starting gear much later, though ... as in 65-odd levels later, where you run into an "elf enthusiast" in the Howling Fjord who collects their clothing.
EVE Online has the starter Frigates you get at the beginning of the game. They're not the worst ships in the game (that would be the Shuttles), but they are a lot worse than the most basic of the "real" Frigates.
The City of Heroes tutorials give characters two level 1 damage enhancements (basically worthless, intended to teach about slotting and combining enhancements) and two inspirations (actually from the top tier of inspirations that won't start dropping until much later). All are completely unnecessary to pass through the tutorial and some players hold on to them to sell afterward to get started buying decent equipment.
League of Legends has Doran's Ring, Doran's Shield and Doran's Blade, items available at the shop for 475 gold (exactly the universal starting allowance) with stats generally helpful for magic, defense, and attack respectively. They're actually fantastic value for money, but severely hampered by the six-item limit. Some players buy nothing but Doran's items to give themselves an early edge before selling them as they get money for more effective items to take their place.
Wurm Online gives you a quite generous set of gear by the standards of MMOs, balanced by the fact that unless you've exchanged a few bucks for in-game currency, it's going to be a long time before you get anything better. It also can't be upgraded like stuff you make or buy yourself; loss of quality from wear and tear is permanent. On the plus side, unlike everything else, it stays in your inventory after you die.
Kingdom of Loathing gives you starter equipment in line with your class and the game's wacky nature: a club for a seal clubber, a stolen accordion for an accordion thief, etc. The equipment is adequate for the low levels, but you'll ditch it at the first opportunity. However, your starter weapon is later required to craft an ultimate weapon for your class. Fortunately, there is a location accessible right from the start — road with craters in the forest — where you can pick starter equipment off dead bodies.
Final Fantasy XIV gives you the basic level 1 weapon of your starting class, and a set of clothes (not armour, not robes, clothes) depending on your race and gender.
Final Fantasy VII gives each character their signature weapon as their starting equipment. They're all the weakest equipment in the game, but in the case of Cloud and Barret, they can't be sold since they're the iconic weapon for that character (and thus appear in cutscenes).
Final Fantasy X Tidus is given a long sword before his first battle. This weapon has 0 customization slots and no bonuses, meaning that every subsequent weapon for him is better. While you can make a long sword via drops and customization it will always have 1 slot rather than 0.
In Final Fantasy XII Vaan and Basch's default swords cannot be bought or found. Vaan's can be dropped by Omega Mark XII.
The Elder Scrolls series you usually go through the "pick up starting equipment through the tutorial" version. However in Oblivion you start off equipped with the unique "wrist irons" item (which are useful since they're weightless and can be enchanted later), and the "rusted iron" and "rough leather" (essentially and even worse version of the basic Iron and Leather items) equipment from the tutorial dungeon can't be found anywhere else.
Rune Factory 3 has a slight twist. When you have your first battle, a weaponsmith lets you choose which starter weapon you get.
In Fallout 3 you get a BB gun in the tutorial. When the game starts proper you're also given a basic pistol and can pick up the aformentioned BB Gun and a baseball bat (if you prefer melee weapons) from your room before you leave.
The starter Pokémon count in that you can't find them anywhere else and they're all pretty basic for their types. All the games also start you off with some of the most basic type of Pokeball (although you'll be buying them for a while at least anyway) and a Potion (which needs to be withdrawn from the player's item storage PC in some games).
In Neverwinter Nights 2, every class starts with nothing but basic unenchanted armor of the heaviest type a character of this class can equip of level 1. The Favored Soul class, added in Mask of the Betrayer but available for the vanilla campaign too, starts with a unique, fancy-looking suit of armor named "Favored Soul's Outfit"... which has exactly the same stats as regular chainmail, and thus quickly gets obsolete.
Each of Dark Souls ten classes has their own unique starter set, with some of them having otherwise unobtainable items. Because of Magikarp Power, a good portion of those items are feasible endgame equipment.
Thieves start with the Master Key, which opens certain doors that would otherwise require a key and some that have no proper key. This can also be chosen as a 'gift' by other classes. Other gifts include the unique 'Old Witch's Ring' (which allows you to talk to the Daughter of Chaos that leads the Chaos Covenant), a ring that grants a small increase in HP, and the Pendant that does absolutely nothing.
Poor Diablo II characters, they don't get any armor, just a class-appropriate weapon: the Barbarian, a hand axe; the Paladin, a short sword, the Amazon, a stack of javelins, the Sorceress, a staff of +1 Fire Bolt, the Necromancer, a wand of +1 Summon Skeleton, the Assassin, a katar, and the Druid, a club. Some of them also get a buckler. It should be noted, however, that these items are flagged as being Starter items, which means they always cost exactly 1 gold to repair. Not that it helps.
Persona games generally start a character off with their school uniform or other everyday clothes and a weak (possibly improvised) weapon scrounged up somewhere - stolen from a school club, provided by your mysterious benefactors, whatever. Persona 4, for example, starts you out with a cheap golf club Yosuke picked up after the replica katana he first offered got confiscated by the cops. Persona has the characters fight through the first dungeon using whatever they could find in a standard hospital room (aside from the one guy carrying an axe).
Planescape: Torment starts you off with a scalpel stolen from a nearby table and nothing else, while your first party member is a floating skull who only has the teeth he originally died with. Fortunately you can replace those very soon with a little scrounging and some side quests (yes, even the teeth).
Xenoblade starts Shulk off with the Scrap Sword. The item's description pretty much admits it's a piece of crap he put together himself just to find off any aggressive wildlife, and it gets replaced with the Sword of Plot Advancement before long.
The player begins the original Knights of the Old Republic on their bed, in their underwear. Basic gear (including a gun, a sword, some clothes, and a couple of medpacs) is stashed in your locker, as you would expect.
In Mass Effect 1, the player is dropped off the Normandy in the first level with a set of four level 1 weapons and armor for each party member. Each other recruited party member would have exactly the same equipment, regardless of their level or where they were encountered. In Mass Effect 2, Shepard wakes up sans weapon and armor in a medical lab and has to grab them from a nearby locker; after the prologue is over they pick up a full set of guns before the first proper mission. Mass Effect 3 is similar; Anderson hands you your first pistol when things go wrong at your trial, and you pick up the rest over the course of Priority: Mars. (In the latter case, it was actually possible for some of the basic guns to be Lost Forever, although it wasn't hard to find replacements.)
The first (real) weapon you start out with in Star Ocean: Till the End of Time is a simple sword synthesized from the rescue pod's computer, which is necessary because you crash-landed in a backwater planet, and convenient for Fayt since he knows how to swing it around by playing virtual games. The actual first weapon you get is a Metal Pipe, an Improvised Weapon which Fayt picked up to defend himself from enemy drones. One of the Battle Trophies requires you to beat the final boss with said Metal Pipe, and the second-to-last Bonus Boss, a feat made possible thanks to the game's Synthesis system.
Starcraft II Wings Of Liberty. The first three missions give you exactly one offensive unit (the Marine), in addition to introducing workers, Medics, and the various features of your base.
Mount & Blade has starting equipment determined via background, but all of it is of poor quality except for the Spirited Corsair, the 2nd fastest horse obtainable (with the first being insanely expensive) being useful depending on your build.
Newly created characters in the Disgaea series always have the weakest weapon of the type their class uses (or one of them) and no armour. Some games in the series don't even give you that, although story characters always have decent items equipped.
In X-COM your first base starts with a little bit of almost everything you can purchase at the beginning. Depending on your style of play either much or all of it is near-useless. Fortunately, the starting enemies are so weak, you can survive.
In the X-Universe series, each game has several "game starts" which change your starting equipment. Typically, the game gives you either a M5 Scout starship or a TS Small Freighter, a small amount of money, and a few weapons. The first game, X: Beyond the Frontier was infamous for its pathetic starting equipment - no weapons, next to no shields, and most annoyingly, no singularity engine time accelerator to make the long travel times bearable. Later games offer more starting equipment, such as X3: Terran Conflict having two starts with a M4+ heavy interceptor and a good array of weapons.