In a game, there are typically two ways the player can become more powerful; they can level up by gaining Experience Points in a Class and Level System or they can get better equipment and/or items which permanently enhance their stats. This trope is about games which only use the latter methods.
This trope typically occurs in more action-oriented games, where the focus is less on character building (since this is one of things which defines the RPG genre examples which are purely this trope are rare) and more on exploration or combat, making Stat Grinding less attractive as a gameplay element (since it keeps the player from progressing or breaks up the action). Exploration based games (particularly in the Metroidvania subgenre, although some also use RPG Elements) will often combine this trope with Utility Weapon; expecting the player to use their new abilities to get to the next set, while combat based games will simply make the player more powerful. Since items can be given and taken away freely, it also allows the game to make the player less powerful (although a good designer won't abuse this). If the items can be picked up in any order (and they're balanced, rather than offering a simple increase in power), it can also make the game less linear. Finding an extra Heart Container is also a good way to reward the player for exploring.
Note that this can still apply to games where the player has some level based progression if there are parts of the game where that's switched off, making items or Heart Containers the only way to increase in power quickly or if leveling up only grants access to more powerful items. Partial examples should only be listed if item based progression is equal to or more important than experience based leveling.
Compare the Sword of Plot Advancement (which progresses the plot rather than the character) and Standard FPS Guns (which tend to follow this trope). Contrast Stat Grinding (where skills advance when the player uses them, rather than advancing for the player to use them).
Cave Story, another Metroidvania style game has you collect various guns, each one leveling up and down individually as you collect XP triangles and take damage.
The Assassin's Creed games generally follow this format, in that you get better weapons to do more damage, and better armor to get more health (the first had a "sync" bar that increased through the game as you did various sidequests).
In Mega Man the entire point of the series was to acquire new weapons from one boss to take down the next. Since you could fight them in any order, the trick was discovering the optimum sequence to fight them in.
Larry And The Gnomes has no leveling system. The player must find projectiles and weapons to get more powerful. Part of the strategy consists in knowing which weapon will be better in the long term, or knowing which will be the most useful for the current stage.
While Ōkami and its sequel Ōkamiden have a leveling system, it is independent from its combat system. The brush techniques may not be equipment, but they do add new powers, allowing for further exploration and more effective combat, as well as some puzzle-solving.
While you do find health upgrades in Shining Wisdom the only way to increase your attack power/skills if to find new weapons.
First Person Shooter
The entire First-Person Shooter genre in general (although less so with more recent games) has this trope as one of its main features (since players are expected to get bigger and better weapons as the game goes on), although many have RPG or Point Build System based progression as well.
Spelunky: You find equipment and upgrades, but you can also get them by going to a shop or performing a ritual sacrifice. The game is randomly generated, and if you don't find things early, the later levels might be too hard to get through.
In Brogue, all of your progression is item-based: Life potions both heal and increase maximum hp, enchantment scrolls increase the effectiveness of any items you use them on, and strength potions allow you to wield heavier weapons for more damage. There used to be an experience and level system, but not in the most recent versions.
In Chrono Cross, you only level up by defeating bosses, not mooks, preventing you from Stat Grinding. The only way to get stronger is to forge stronger weapons/equipments.
In Minecraft your character's baseline health and physical abilities never change. You can gain levels of experience, but you spend them to enchant equipment. Your strength, health and ability to interact with or shape your environment entirely depends on the type and quality of the items you own or use.
Vagrant Story; Ashley's stats can be permanently boosted by killing bosses or finding rare elixirs and wines, but the game's mechanics and Random Drop rules encourage the player to make Ashley stronger by finding new weapons and armor, or by reforging and improving existing ones in workshops.
Monster Hunter runs on this trope as well. Your power is based completely on the equipment you own, and the point of the game revolves almost entirely around finding new ways to create new gear. You hunt monsters to skin them for body parts, of course, but you also have to mine for ores, catch fish and bugs, collect herbs and mushrooms and plant crops to really get the most out of everything.
In Dark Cloud, you can increase your characters' Health and Thirst meters, but you can only increase damage by adding enhancements to their weapons and, eventually, building them up into bigger weapons. You do find new weapons occasionally, but after you've built up one weapon a couple times, even the mightiest new weapons start to pale.
In Mega Man Battle Network, the only stats that you can find increasers for are Hit Points and kB (the latter is used to set a chip as available first turn). Because of this, the only real way you can increase your power is by finding and equipping more powerful chips and Customizer parts (which can increase other stats, like rate of buster fire, and give other abilities).
While multiplayer characters in Mass Effect 3 do gain experience and upgrade their skills, their levels are capped at 20 (single-player cap is 60) and the players have to reset them back to level 1 if they want to keep advancing up the score ladder. This effectively renders character progression meaningless in a long term, while rare weapons, weapon upgrades, and advanced equipment mods (randomly awarded for winning online matches) become crucial on higher difficulty settings.
Any of the Record Of Agarest War games are made out of this trope. Sure you level up and such, but most of the time you'll be doing a lot of equipment customization.
The concept overlaps with And Your Reward Is Clothes in The Elder Scrolls III: Morrowind. After a certain point, the rewards for the main quest (and some other quest lines) become things you can wear, intended to be symbolic of recognition as a great war-leader, only occasionally with a marginally decent enchantment or other bonus.
God Eater Burst relies solely upon equipment to increase your abilities. The only difference between a character five hours into the game and fifty hours in is equipment load-out, and you can always opt to go into the most difficult fights with the most pathetic gear if you're feeling masochistic. Party members are static as well unless the story changes their gear or tactics.
Third Person Shooter
The Tomb Raider games follow this trope, giving you more powerful weapons as you progress whilst all other skills remain static from the start.
The Jak and Daxter series gives you new abilities and weapon upgrades from the second game onwards (the first game just placed temporary powerups in certain parts of the levels) as you reach certain points in the story.
Many MMORPG games become this trope in the end game, when the players have reached the level cap, so their only means of progression is through better equipment.
Realm of the Mad God is intentionally designed such that it is easy to reach the max level in less than a few hours, then it becomes this trope. While leveling up does boost your stats, the main focus is in fighting monsters to acquire better and better equipment. There are no prerequisites to use any of the equipment (besides obtaining the item).
In Spiral Knights, the players don't gain experience themselves, instead, you can craft different pieces of armor and weapons, each one with its own elemental bonuses and weaknesses, and each item requires Experience points (or "Heat") separately, and usually has an additional bonus when they reach its level cap. They (and the levels' difficulty) are also divided on tiers. In order to be allowed to go deeper on the clockworks, you also need to craft better equipment(usually using a level capped item of a lower tier).
Dynasty Warriors Online. Oh yes. The only stats you get at all are never attached to your character. "Progression" isn't the main idea, it's just being stronger during battles, but the weapon you use makes up 95% of your combat stats. This dips into the bad end when all weapons have unique movesets but the stats are constant as well, and it in fact rendered one of the fan favorite movesets obsolete because of the poor stat choice, but other than that it works fairly well.
The whole "catapult" genre of flash games such as Toss The Turtle has this as the main game mechanic. The farther you launch the projectile the more money is available for upgrades, and each upgrade helps you go further, and so on ad infinitum.
Paper Mario handles parts of the progression this way. While you can level up your HP, FP (Mana), and BP (Badge Points, which define how much you can equip), attack and defense are solely dependant on your equipment. Mario's jump and hammer become more powerful once he upgrades his boots or hammer, which happens at certain points in the game, and everything else is dependent on which badges he has equipped.
Non-Video Game Examples:
Kingdom Death is a game based around using boss parts to craft armor and weapons. Since there's no leveling system, this means that all character progression is based on grinding out armor.
In Shadowrun character progression through equipment/money (gear) and experience (karma) are equally important. There's actually an exchange rate that Game Masters can use as a guide to convert between the two when deciding how to reward players for their quests. This is unusual in table top games in that gear is a formalized part of the progression system, with the same importance as skills.
In the Noob franchise, quite a few characters are already level 100 and hence implied to dbe doing this. The Justice guild main roster is actually seen renewing a piece of equipment or another from time to time.