In Warhammer, there are a number of units that will have the player using them laughing in glee as the opposing player growls in frustration. The Lizardmen have Skink skirmishers— small and skittish blowpipe users with Poisoned Weapons that love hiding in terrain and can shoot in any direction.
Conversely, in Warhammer 40,000, this line of thinking is what lies behind the so-called "Tarpit" units. Tarpit units are essentially cheap, expendable foot-soldiers that are used to swamp a powerful unit in close combat. They probably won't deal much damage, in fact they'll likely get their asses kicked if they're tying up anything worth tying up, but they're just meant to tie up the enemy unit, taking so much time and effort to kill because of many wounds, many models or both. Some classic Tarpit units include Boyz or Gretchin for Orks, Cultists for Chaos, Ripper Swarms or Gaunts for Tyranids, Conscripts for Imperial Guard... In fact the only armies that don't really do Tarpits are Eldar and Dark Eldar, which is hardly surprising considering their whole army dynamic.
Pugwumpi/jackalrats in Pathfinder: tiny, very squishy humanoid fey with just enough damage reduction and spell resistance to be annoying at the low levels in which they're normally encountered, and no particular offensive prowess besides the mind-bogglingly annoying "bad luck" ability that, in game mechanics terms, forces players to roll their dice twice and use the lowest result. Since low-level parties are particularly dependent on the grace of the Random Number God to get anything done, this can make fights against the little wretches annoyingly drawn-out, despite the fact that a solid blow from even a low-level fighter could kill them.
Battletech has Light mechs, light and medium vehicles, light Battle Armor and infantry. The average heavy mech costs between 1500 and 2000 Battle Value. You can get between three and eight light mechs for that, between five and twelve light vehicles, between ten and thirty light battle armor, or between fifteen and sixty infantry units. In infantry's case, this becomes a major time consuming play by your opponent, because A) Weapons that are good vs. Infantry are rarely of any value vs. anything else, so relatively few mechs are equipped to face infantry; B) Even if you manage to kill 2 platoons per turn (optimistic goal), it will still take you 20+ rounds to clear them all. The only saving grace is they are slow and short-ranged (except for field guns). Used properly they can be dangerous against an unprepared or reckless opponent, but most of the time they are time-consuming cannon fodder.
Kobolds in 4th edition Dungeons & Dragons have shades of this, as their Shifty ability means they can move in, take a swipe at a target, then dart back away, forcing the PCs to come at them if they want to get a swing in (often into an ambush, trap, or difficult terrain). Of course, Kobolds in D&D have always had this sort of characteristic, especially if the DM plays up their love of traps. And then there are Tucker's Kobolds...
The pawns in Chess are the weakest pieces in the game, but there are four times as many of them as any other piece in the game. If your opponent has any skill their pawns will most likely block or protect almost every single strategic location or enemy piece you can find. Even their 1 on 1 weakness against other pieces can work in their favor, since sacrificing anything other than one of your own pawns to get rid of one of the pests will almost always worsen your position in the game. This is fully acknowledged in the Meta Game, where the pawns, as a collective, are considered the most important pieces in the game after the king himself.
François-André Danican Philidor, commonly regarded as the best chess player of the 18th century, is best known for writing "Pawns are the soul of chess", a maxim still well-known to all chess players.