This is basically what the "weenie horde" deck archetype in Magic: The Gathering is all about. The inherent problems with it are the relative frailty of small cheap creatures and the one-draw-per-turn bottleneck that ultimately limits the size of the horde one can muster; good weenie deck designs generally include cards that deal with both. Screaming "elf deck!" can scare MtG players not prepared for weenies nearly as well as screaming "zerglings!" can RTS players not adept at dealing with rushes.
Saprolings are the prime example of a weenie horde, although usually a slower one. There is a deck that involves a combination of cards allowing the player to generate a literally infinite amount of saprolings creatures which run over any opponent in short order.
This differs from classic Zerg rushing in that Myr Enforcers, the deck's namesake Arcbound Ravager, and other creatures common in Affinity aren't weak at all, and aren't "really" cheap—they're just easy to make inexpensive. Also, Ravager Affinity is a Genre Savvy use of Conservation of Ninjutsu, with the basic early rush tactic compounded by saccing all the artifacts to power the Ravager when it comes on board. The main trick of Ravager Affinity is to force at least one attacker through unblocked by the Zerg Rushing, then feed the Ravager and cannibalize it to transfer the + 1/+ 1 counters to the unblocked attacker. Several cards that made the deck work were quickly banned, most infamously Skullclamp.
Another classic example of a rushing archetype is Sligh. Variants of this deck play small red creatures - generally among the worst in the game compared to other colors - and smack them into the opponent as quickly as possible, then finish the opponent with fire. Despite the relative poor quality of small red creatures, mono-red Sligh and other similar decks using different balances of creatures to fire spells appear in every tournament, and their speed is the standard against which all other decks are measured in Magic: The Gathering's Meta Game.
The Onslaught Block enhanced the Goblin tribe until the strongest Goblin decks superseded classic Sligh.
Somewhat closer thematically to the Tyranid example below are the slivers, in both the actual game and the world the game takes place on.
The generally-recognised ultimate sliver combo, which effectively allows an infinite number of sliver tokens in one turn, is that of Heartstone (or Training Grounds), Ashnod's Altar (or Basal Sliver), and Sliver Queen. Spawn a token from the Queen, sack it using the Altar or to itself using the Basal ability, then produce two more with the bonus from the Heartstone or Grounds, and repeat until your opponent forfeits. A green-heavy Sliver deck depending on a Aluren and Overrun combo can put a very large number of Slivers on the table within a single turn; but has the weakness of requiring a large amount of mana to use, and is therefore slower to activate than a true Zerg Rush.
The card Coat of Arms is seemingly designed especially for the Zerg rush tactic, as it gives +1/+1 to all creatures for those of the same type(s) in play.
When it comes to modern M:tG decks, most "Zerg Rush" tactics - if they can really be called that outside of the weenie deck archetype - basically boil down to perpetual motion systems, since the developers try to avoid letting you win without thinking. Example: the above mentioned Ashnod's Altar gives you 2 mana when you sacrifice a creature to it. Myr Retrievers cost 2 mana to summon, but they pull any artifact out of the graveyard the moment they die, and since they count as artifacts, they can resurrect each other. A Cloud Key can and/or anEtherium Sculptor always does reduce the cost of artifact spells. With an extra Myr Retriever in the mix of this combination, one literally has an endless supply of mana (sac Myr to Altar = 2 mana; Myr is dead, get other Myr; summon Myr for 1 mana; 1 mana left over). Add this to an artifact or creature that spawns 1/1 counter creatures and you instantly have an endless Zerg Rush powered by infinite resources, and Kerrigan would be proud. This combo also works with "plink cards" that cause a single point of direct damage for the cost of one mana, making this a painful way to die.
Woe to those who allow the use of Unhinged cards in their match and get slapped with Ashnod's Coupon, only to come back and see about six Cheatyfaces on the field.
Selesnya and Boros both dabble in this in Return To Ravnica. Selesnya has a heavy focus on creature tokens, with their Guild Ability, Populate, effectively giving you a free token of an existing one on top of whatever was the effect of the card you were casting. They also have tokens ranging from small songbirds, to gigantic militarized wurms, to giant sentient temples. Boros on the other hand focuses on small, efficient creatures. Any tokens they generate instead are simply weak footsoldiers, and not really on the scale Selesnya can produce. Boros, however, have their Guild Ability (Battalion) dependant on massive amounts of attackers to get disproportionally better effects; ranging from dealing direct damage, making your troops invulnerable, bestowing any number of special effects on all existing troops, or just make a singular troop really, really strong.
Yu-Gi-Oh! has an Archtype that plays to this tactic, technically speaking. The Blackwings (Black Feathers in Japan) specialize in swarming the field with monsters that have effects that play off each other can can One Turn Kill pretty quickly. Not really a "Zerg Rush" due to the 5 monster limit the game has, but it's pretty close.
A better example would be the main strategy of an Infernity Deck. Though it's more of an example of the scale from this into a Boss Rush, which is a Zerg Rush consisting of the biggest monsters you could ever summon in one turn. To point, Infernity Beetle (a level 2 monster) and Infernity Daemon/Archfiend (a Level 4 monster) together in the right combination can result in 5 and more synchro monsters, especially Trishula. This was such a devastating strategy that the succeeding banlist had to target the already expensive cards, making the key monsters R1.
The short lived Kingdom Hearts CCG is a huge example of this, if a Light Deck has a bunch of low level friends (0's and 1's, for instance), or any Dark Deck, which can basically get up to level 8 friends in a few turns if they can, it doesn't help that the only Level 9 Dark Card (Dragon Maleficent) allows you to discard cards from your hand to lower her level and play her sooner, meaning, if you decide to play her with a full hand of six cards, you can discard the other five cards and make her a level 4 Dark Card. Then you can start bringing out big ones that block your opponent from playing friends, (Captain Hook prevents Peter Pan and Tinkerbell from being played, Oogie Boogie prevents Jack Skellington, Maleficent prevents Beast, and so on). Or if you really want to be a douche, play Darkside, which removes all Level 1 and 0 friend cards in play from every player's (up to four) playfield.
In the Deadlands CCG Doomtown, lots of cheap dudes + We Got Ya Surrounded = big shootout bonus (and they're easier to heal in case the bonus still isn't enough). The original Blackjacks home is good for this; the more Blackjacks dudes (or Deputized drifters) you have, the more you can boot to rob ghost rock and buy more dudes.
The Naruto CCG has the card "Power of the Youth", which lets you send out as many teams as you want as long as all your in-play Ninjas have an entrance cost of 3 or less. Combine this with cards like Idate, [SUPERHUMAN SPEED] effect Haku, and [PARTNER] effect Akamaru, and you can quickly build up your own little army and swarm the opponent before the gamebreaker ninjas start showing up. Did I mention this card lets you Zerg rush for 2 turns straight?
In Smash Up, this is the Robot faction's hat. Having only 2 Action cards but 18 minionsnote Most other factions have 10 of each, they focus on drawing masses of weak minions to capture bases. They have minions that summon more minions, minions that can destroy stronger foes if the player has many minions on a base, minions that power up others, and one that is powered up by having minions in play. Needless to say, Robots are great at dropping off multiple units on a base in one turn and potentially strengthening them.
The Zombie faction. They have abilities that that put large numbers of cards from the deck/hand into their discard pile, and abilities that draw cards from their discard pile, allowing them to easily respawn large numbers of minions that have been killed off (by opponents or the player themself).
Since Smash-Up requires the player to combine two factions together, a combination of Zombies with Robots gives a massive Zerg Rush of minions that just won't stay down.
In the Cthulhu Mythos-themed expansion, the Innsmouth faction uses a variant of this strategy - it's very easy for a player to draw a slew of weak minions, but playing them isn't as easy.