Warhammer 40000: Tyranids all over. Also a fairly valid tactic for Orks, Imperial Guard and Chaos cultists/mutants/plague zombies, too. (Note that of the above mentioned factions, the Imperial Guard is the only one without a hive mind, berserker bloodlust, mind control, or all of the above. Needless to say, their morale is not good.)
Of course the Guard do have an advantage the others don't: damn good artillery, and lots of it.
The Imperial Guard really takes the cake for Zerging, as they can also do this with tanks as well. If any other given faction mechanized their entire force, they would have 17 Tanks in total. The Imperial Guard, however, can come at a whopping 59 Tanks (41 Transports + 6 squadrons of 3) based on their legal Force Organisation chart, in part due to the fact that they alone are allowed tanks in squadrons and have the Platoon System, which gives them a lot of squads for one actual choice (each squad can take its own transport).
The 'nids really do take the cake here; their main soldier, the 'gaunts' (in all forms), main job is to rush at defences so they'll run out of ammo when the bigger 'nids show up; most of the time it takes weeks of non-stop rushing, and in fact most 'nids don't even have a digestion system. The main reason why they do this is because since the Hive Fleet eats everything, they eat their dead (and their still living forces) so it's still a net gain even if they lose billions.
'Gaunts can be returned to the table every time they are killed to represent the endless swarms of them that exist (with the beautiful ability name of Without Number). Apocalypse battles (very big games with lots of 'gaunts) actually have a rule where opponent's troops will run out of ammo if they shoot too many of them.
See the Futurama example here? With the ships clogging the enemy's main cannons with their wreckage? The Tyranids did that to Tyran, their Trope Namer planet, using their own bodies.
Units that are classified as "Swarms" are literally a swarm of creatures too tiny to count as individual combatants, but nonetheless can fight as a cohesive unit (which can be then formed into squads). They are weaker than even the most basic troopers in the army, rarely have ranged weapons, and can even be totally ignored when shooting in favor of larger targets (normally you'd have to shoot the closest thing possible). Their use? They have an ridiculous amount of Wounds (HP) and can tie up enemies in combat, so they can't do squat until your big bouncers come and mop them up. The Wounds-to-Price Ratio on these are also the best, so you can get a literal meatshield for dirt cheap.
And as per 40K fashion, even these were turned up to eleven in the 5th edition. Swarms now can provide Cover Saves to units behind them, meaning that other troopers can literally use them as moving shields that would otherwise be only attainable through being in cover. Almost all Swarms are also buffed with poisoned weapons and, in the case of Nurgle's Nurglings when Epidemus is around, power weapons. They are still as dirt cheap as always.
With the new Necron codex, the Scarab Swarms are quickly approaching Lethal Joke Character status. They are the sole swarms equipped with an anti-tank weapon, which reduces the armor of the tank by 1 for every hit in close combat and destroys the tank if one of the armor ratings goes down to zero. Given that swarms possess a significant amount of attacks, just a handful of scarab swarms on a tank (worse if the tank stood still the last turn) can completely devour it in a single turn, even if it's something hardy like the Leman Russ.
Imperial Guard special character Kubrik Chenkov has a rule named "Send in the Next Wave." Any squad of Conscripts can take this upgrade to allow them to return to the table as a fresh unit at the beginning of the player's turn. An army can have up to six squads of Conscripts, and each squad can have up to fifty men. Since they're troops, they can hold and contest objectives. Fluff wise, Chenkov's regiment has been refounded an innumerable amount of times because he's reportedly killed more of his men than his enemies have combined.
Chenkov once infamously ordered the construction of a wall to be made hastily in the field. When they ran out of bricks and mortar, Chenkov ordered a whole platoon of his to be executed, their bodies used to "fill in the gaps" so to speak.
Warhammer Fantasy Battles also has notable examples of Zerg Rush applying armies, such as the Skaven, Orcs and (mostly) Goblins and the Gnoblars (the latter only existing as an army list published in Games Workshop's "White Dwarf" magazine).
In the 8th edition the rules actually allowed for such tactics to work, and possibly quite well. In previous editions only units in the front and second rank can fight, and only the front can ever fight at full capacity (the second rank was only allowed to use spears and make one attack). Most ranks are only 5 models strong, but units could usually number 20, so this means that only ever 10 models got to fight regardless of how big the unit was. Newer rules introduced the "Horde" rule, where models had to make ranks of 10, and up to three ranks can fight, with the first two fighting at max capacity. This meant that a total of 30 models can fight, three times as much as before.
Bretonnians take this concept and turn it up to 11 (or 12, to be precise) where their Lance Formation allowed them to Zerg Rush with armored knights on barded warhorses.
Skaven go over 9000 with the concept as they literally have the cheapest troops in any army (of both game systems) and can fire into combat. For 1000 points, they can field 500 models. Although whether you can is another matter.
The Soviet Union's primary strategy in Axis And Allies is to build half a dozen infantry or more each turn. However they are only good for defense, Germany has enough industrial capacity to do this with tanks.
The standard method of attack of Dungeons & Dragons kobolds when they're out of their mines. Defensively, though, the entire point of kobolds is to lead opponents through deadly traps and mazes, and areas where they have the edge (e.g., leading medium sized creatures into kobold sized spaces). They only attack en-masse when there is no other option (i.e., when they're not on turf they can make use of in this fashion). This is expanded on in Races of the Dragon, the 3.5 supplemental book covering kobolds and other races descended from dragons — kobolds take a very collectivist view to life, being extremely Lawful Evil (leaning Lawful Neutral) — while the first defense of a kobold city is going to be its hundreds of extremely deadly traps, ambushes, etc., as a last resort, the kobold men will throw themselves en masse at attackers to buy the women and children time to escape with the eggs, the idea being that their lives are a small price to pay to ensure the survival of the town.
According to Moneyball, a player of tabletop naval combat simulations did this with PT boats and won every round in a particular tournament, although most of the boats were destroyed each round. The next year the rules were modified to emphasize mobility, which PT boats lose when damaged—so he scuttled every boat as soon as it took damage.
In BattleTech, this was part of the in universe purpose of the Protomech, which was smaller and cheaper than standard Battlemechs while faster and more heavily armed than Battle Armor, allowing them to be built and fielded in large numbers. In gameplay, it's sometimes invoked by flooding the map with dozens of super-cheap infantry or light vehicles, though this is typically considered to be extremely bad sportsmanship due to the Game Breaker nature of the tactic.
In Strike Legion, this is the Imperium's tried and true tactic. With three million systems under their control, a fleet of fifty million warships, and countless trillions of bodies to throw at the target, they've been wearing down the vastly smaller but vastly more-advanced and better-trained Star Republic through raw numbers. One quote from the Empress herself casually has her order an additional million ships to the front line. At the same time, the Empress has recognized that relying purely on sheer numbers is still inefficient, and has ramped up production of more advanced ships, frames, and Super Soldiers to match the Republic in quality as well as quantity.
Even with all the powers of Caine on their side, the Vampires of World of Darkness know if the masquerade were ever broken and humanity learns of their existence, they would be wiped out in no time.