In D Gray Man the Millennium Earl launched a siege on the Exorcist's base to retrieve a bit of Applied Phlebotinum from them. It consisted of vast numbers of Akuma, which most Exorcists are capable of dispatching with relative ease. It nearly worked too.
In Mazinger Z, often Baron Ashura sent large numbers or Iron Masks to attempt to murder Kouji Kabuto -and his friends- when he was out of Mazinger-Z or to try to overwhelm or get Mazinger-Z and Aphrodite-A confused (and it worked with Aphrodite once. They were so many, so fast and so annoying they got her distracted and slowed her down). Curiously, it was inverted in the original manga once: in one of the first chapters three Iron Masks crawled in Kouji and Shiro's home and they almost got Kouji killed.
In the original manga Dr. Hell also used that tactic in the Final Battle, mobilizing two of his armies, a Cool Ship, a Cool Airship and forty Mechanical Beasts against Japan and the Mazinger army. In Mazinkaiser Dr. Hell also used twice that tactic (and it worked in one of them!) and The Mykene Empire sent huge numbers of Warbeasts all over the world to conquer the surface. Similarly, the Vegans from UFO Robo Grendizer attacked Earth cities with vast fleets of mini-ufos, often backed by at least one mothership and several Saucer Beasts.
This was part of the defense of One Piece's Enies Lobby. A 10,000 man force of Mook Marines and other forces set to defend against any attempt to attack against it. While Luffy single handedly defeated a good tenth of that force and more fell to the allies the Straw Hats brought with them, the power of the Zerg Rush kicked in after the Straw Hats got to where their battles with CP9 would take place and their allies were subdued and captured.
Don Krieg and his pirate armada presumably utilized this tactic as well. While they call themselves the strongest pirates in the East Blue, Luffy informs them that they're just the one with the most people.
Fifty ships. 5000 men. East Blue calls it an invasion force. The Grand Line (Mihawk in particular) calls it target practice.
This tactic is pretty much useless now against the protagonist and in the majority of the New World. The ability "Conquerer's Haki" allows the user to impose their will on someone, and, if the target's will is weak enough, knock them out. Luffy used this ability to knock out 50,000Mooks in one blast - and the only reason he didn't knock out the other 50,000 is because his crew needed someone to fight.
When he begins to seriously utilize them for scouting is after he realizes that any knowledge the clones gain is assimilated back into himself when he dismisses them. Along with his unusually-high chakra this allows him to summon entire fields of himself to all train at the same time - enabling him to do so hundreds if not thousands of times faster than a regular person.
And don't forget Madara's one hundred thousand Zetsu plant-men army made with First Hokage's chakra
The samurai of the Land of Iron charge Sasuke when he threatens the meeting of the Five Kage. Of course, Sasuke has finally become so enraged with everything that's happened, that he hacks his way through a good half dozen of these men. In hindsight, he gave them a chance to back off.
The Safeguard in Blame!!, since they have many a Mook Maker on hand and act as little more than a kind of elaborate anti-virus system (if the virus was humans).
Mobile Suit Gundam 00: Gundam 00 seems to have been plagued by this one because in the series, a lot of Devines and Brings and also the ELS in the movie.
To expound: in the beginning, the only way Earth felt it had any chance of defeating the Gundams of Celestial Being is by overwhelming them with numbers (with varying degrees of success). They were that powerful by comparison. Once everyone has access to GN technology, Celestial Being had to focus better on tactics than just being a few overwhelming powerhouse units to deal with the same mass of forces.
Played straight in the last battle. Ribbons attacks the combined forces of Celestial Being, the A-Law group that defected, Katharon, etc.. with a continuous swarm of Trans-Am-equipped Gaga units. They do little more than activate Trans-Am and kamikaze everything.
This is also the main tactic of the ELS, most of which don't even have projectile weapons (until the final battle, in which they begin copying human weapons), but there are just so many of them that it's nearly impossible to shoot them all. One character in the final battle bemoans that the human forces are outnumbered over 10,000 to one... and it's not clear whether this is intended to be hyperbole.
A non-violent version appears at the climax of Durarara!!'s first season, when Mikado Ryuugamine publically confronts Namie Yagiri in a crowded hotspot of Tokyo nightlife: admitting he has neither "the power nor the wisdom" to reason through her delusion, he will instead rely on numbers, and presses the "Send" button on his cell phone. Every phone for at least a city block goes off at once with an incoming text, simultaneously revealing every single person present to be a member of the Dollars, and Mikado himself to be their mysterious, urban-legendary founder, intimidating Namie and her mere handful of armed goons into retreating. Well, okay, Celty driving down the side of a building and going berserk may have helped that along a bit, but still.
The climax of Pokémon: Destiny Deoxys has the security system of LaRousse City going haywire after Rayquaza's rampage on the city. The security robots go and attack Rayquaza (as well as the two Deoxys when they attempt to protect Rayquaza) en masse until Ash shuts down the system.
Used in Baka To Test To Shoukanjuu, Class F sets up traps so that they can overwhelm Class A students one at a time. Pretty effective until the rest of Class A breaks down the desk barrier Akihisa set up.
In Pokémon Special, a whole bunch of Unown that the Sinnoh trio saved in a previous occasion bombard the Galactic grunts blocking the entrance to the Veilstone Galactic Building so that Pearl can sneak in unnoticed.
The aliens in both Gunparade March anime series follow this trope only by virtue of their superior numbers and mindlessness. Individual units are stronger than even the Humongous Mecha of Earth (let alone more conventional vehicles, which are present in greater numbers) and the latter are forced to survive via better tactics and superweapons. Orchestra does show humans finally catching up with genetically engineered Mons.
The Gunmen of Tengen Toppa Gurren Lagann appear in massive numbers when engaging in large scale combat, at the command of their respective leaders. To a much more frightful degree, the attacking forces of the Anti-Spiral race work in deadly efficiency. Fresh troops come in increasingly greater numbers as fast as they fall, pulling out the wrath of lethal bait-and-switch tactics designed to kill an enemy the moment the underestimate their foe.
In Hellsing Alucard's level 0 releases the sum of all the millions of souls he's collected over the centuries, literally drowning entire armies in them. The irony is that a variation of this trope is The Major's tactic against Alucard. First he let's the vampire battalion murder most of London resulting in above 3 million deaths. Then the Vatican's army arrive and fight the vampire battalion. Alucard shows up and uses his level 0 state to kill them all. The goal is not to hope that his large army will kill off Alucard but to trick him into beliving that this was their plan causing Alucard to belive he had been victorious. This would cause him to start to absorb all of the blood (and with it the souls) of all the millions that have died. Then Schrödinger would kill himself and be absorbed so that Alucard would gain his reality warping powers which would cause Alucard to seize to exist.
In InuYasha, Naraku's minor yokai do this almost every time, with nearly identical results.
Seen in Sword Art Online, when the Aincrad Liberation Front (also known as just the Army) rushes a floor boss with precisely zero preparation. They didn't even scout first. The ones who rescued them are more than a little pissed at their stupidity, and explicitly call them a "zerg-rush guild." It later turns out that there is a political fissure in the guild, and the leader of one faction managed to trick members of the other into using this tactic.