Bats were always irritating, but an update made them harder to hit, and made them living, rather than undead, which allows them to have attributes that make them even better. Not to mention the fact that they replenish their health on successful hits, all of which makes them horrible to kill.
Orcish Assassins have fairly low-damage attacks and are reasonably squishy...but they're annoyingly hard to hit in just about any terrain and their ranged attack mode is both highly accurate in turn and inflicts poison on anyone it hits. Since the usual ways of dealing quickly with evasive units (marksmanship and magic) are also tied to ranged attacks against which the Assassins can retaliate with their poisoned throwing knives, this makes them a considerable pain in the neck despite not being in and of themselves overly lethal.
Ghosts, especially the Wraith and Spectre upgrades, are annoying in much the same way as bats. They are very mobile over almost any terrain and have decent evasion. They also have life-draining attacks and high resistance to four out of six damage types, making them very difficult to kill without the right units.
Anything with a Sleep or Berserk staff in the Fire Emblem series.
Hector mode for the 7th game adds Pegasus Knights to early chapters, before the player has a method of countering them. One annoying one is in chapter 14. If a Pegasus Knight gets low on HP she will flee to a fortress to recover health. While a logical move, this is annoying for 3 reasons: 1. You can't reach said fortress, as it is over water. 2. It is a kill all enemies chapter. 3. If you are going for rankings (where you are ranked on, among other things, the number of turns you take), it will add about 4 turns to your total.
Pegasus Knights in general apply. Due to their high move stat, they can often fly around your tankier units and attack your back row fighters, their higher than normal resistance makes mages less effective against them, and their high speed makes them a pain for slower characters. What stops them from becoming Demonic Spiders like their wyvern-mounted counterparts is mainly their rather low strength and defense, the latter of which is only made worse by their natural bow weakness.
Fire Emblem Fates brings Ninjas to the table. They're fast and hard to hit, and almost always carry weapons that have 1-2 range and, in a series first, debuff your stats. They are often seen alongside harder-hitting units that can exploit your units suddenly having a few points of defense less.
From the same game, there are some reinforcements that appear in the battle, nicknamed Asshole Reinforcements by the community. In Conquest they become even worse - because they don't even give experience yet are just as threatening as any other reinforcement is.
Myrmidons are annoying in most games, since, despite usually having low strength/defense, they have very high speed and skill, making it easy for them to hit, dodge, and double your units. However, they're even more obnoxious in Fates, thanks to the skill Duelist's Blow, which grants them an additional 30% avoid when they initiate the attack. And since most combat in the series happens in the enemy phase... you better hope you have someone strong enough to take them out on your turn.
Shadows of Valentia has Mogalls, which can self-multiply at random and are fast enough to double half your army. Worse yet, they're considered Great Terrors, which means that Expel doesn't work against them. You have to pick them off one by one and hope they don't multiply faster than you can kill them.
In the Nintendo Wars series, foot soldiers can be spammed for the most part — the worst offender by far is the well-known 'mech rush' tactic, using bazooka infantry known as 'mechs' that have the offensive power of a tank and a third of the cost. Not only are they frighteningly effective in large mobs, but they can function as meatshields for range-fire units and only high-cost units are able to insta-kill them (and being mobbed by three of its compatriots, who cost about as much as the unit you used, will usually cripple it). This is, however, averted in Game Boy Wars 3 (where plenty of units can wipe them out in a single attack, including the 3rd cheapest unit in the entire game).
Game Boy Wars 3, however, promotes the Submarine to this status. In this game, they're ranged attackers (plus you can actually move and attack in this game!) that can only be damaged by a grand total of two units; Anti-Sub Helicopters and other Submarines. Also exclusive to GBW3 is unit promotion, and promoted Submarines trade their ranged capabilities for an area-of-effect attack. And they still are only vulnerable to two units.
Seaplanes in Days of Ruin. They are the only unit that can attack every type of unit, are moderately powerful, and have a high movement range (they can be Launched out of carriers that themselves have a relatively high movement range; on the turn they are launched they effectively have a range of 12 spaces). Also seeing as how they are planes, you don't have that many options to deal with them. What holds them back from being Demonic Spiders is that they only get enough ammo for three shots, and run out of fuel relatively quickly.
Dominions 3 has several spells that serve this function. They summon large numbers of 1 Hit Point, fast-moving, hard-to-hit units that do minimal damage but distract enemy armies, who stop to kill them while your troops shoot/cast spells/get into formation. Nothing is more annoying than having your 50-man army swatting dragonflies while arrows rain down on them from the 20 enemy archers who by all rights should be being slaughtered by your infantry.
Trolls from the game Dark Legion aren't very deadly, but have high health, and if facing another unit that is melee only, will wear them down considerably through chip damage. Killing them at long range is just plain annoying shoot-n-scoot...for about three minutes apiece.
Super Robot Wars W has an exceptional number of Super Small-sized units compared to other games, mostly due to the inclusion of Tekkaman Blade and Detonator Orgun, both series with a bent towards Powered Armor. This means lots of very small, very hard-to-hit enemies, such as Sol Tekkamen, Birdmen, and Alien Tekkamen that you usually have to use your Spirit abilities to get rid of. Especially aggravating in the fact that the game freely gives Birdmen to the GaoGaiGar and Full Metal Panic! bad guys to fill out their ranks.
The Victory Gundam units from the same game are also frustratingly thick-skinned, with health above 20,000 and stats similar to the the original mooks.
R-Einst in Super Robot Wars: Original Generation 2. Not horrendously strong, but their high Mobility sucks the SP of large units like Daisanger dry. Especially when you're trying to take them out on a time limit - most of the units that would have trouble hitting them would probably still win in a duel if properly armored, but with OG's fondness for "win in 4 turns"-esque Skill Points...
Also from OG2, the damned Gespents that the Shadow-Mirrors brought with them. The ones that require either Focus and just plain luck, throughout the first half of the game and a bit beyond that point, or Focus -and- Strike(spending even more of your precious SP, yay![/sarcasm]), and then just a bit more luck for their return fire, no matter how good a chance to dodge your own unit has. Large amounts of HP(for so early in the game on 'fodder' units), quite decent armor, and some heavy-hitting weapons. This player has ended up cursing enough to peel paint thanks to the amounts of restarting those bastards have caused...
Bartolls that runs on ODE in OG Gaiden might count as well. Basically, these units have a shared morale pool, and killing one of them INCREASES the morale. The more you kill them, the more you'll need to consume your SP to make them get hit by your attack, and for them to miss their attack. You could try to take them out with area attack maps (like the Cyflash), but... well that'll waste SP. As if there's not enough reason to make the ODE and its creator (Juergen) scrappies...
Alpha Gaiden and 3 both feature the Ghost X-9. In the OVA proper, a single Ghost was a pain in the ass for Isamu and Guld and required a touching Heroic Sacrifice (the movie version made the final defeat more intense and the sacrifice way more graphic). In Alpha Gaiden and Alpha 3, the enemies take it one step further and mass-produce the Ghost. And it is a Small sized unit, meaning that unless you use the 100% hit-rate Strike/Lock-On seishin, chances of actually hitting a Ghost are almost non-existant. To top it off, the Ghosts have high accuracy hit rates on top of their insane evasion stat, meaning that the Ghost X-9 is practically the best example of a Boss in Mook Clothing for the Super Robot Wars series.
And as a more general example, any mook with the Support Defense skill. To clarify : an unit with that skill can Take The Bullet for adjacent units, reducing the damage in the process. The games love giving you large crowds of these protecting each other to fight, resulting in battles that are easy but last for hours unless you MAP attack them to death.
Phantom Brave featured a mushroom that would rip the weapons out of your hands and throw them Out Of Bounds. Forcing you to return to Phantom Isle to re-equip your units.
The little rebel armies that pop up at random in Rome: Total War. All they do is stand there blocking your roads and trade routes, and they're not even hard to kill. The problem is that they will invariably appear in your heartlands, which you have pacified, and where you consequently don't have armies standing around ready to deal with them. You might have such armies for just an occasion, but the net gain from trade is only barely more than you spend on upkeep.
Monsters of the Mothman species in the Disgaea series can be a nuisance due to their unrestricted movement, a high movement range, multiple status inducing attacks, and starting from 3, a base 50% evasion chance against attackers directly adjacent to them.
On some stages, there is the Clone geo effect and/or Mook Maker portals. If you haven't bothered to level a balanced party, you'll quickly flood a stage with more monsters than you can destroy.
Speedboats in Advanced Strategic Command. Very fast, very stealthy, Move After Attack... weakly armed. They can be very dangerous in big swarms and/or using their small transport capability to bring waves of infantry to a shoal, but it's not what will happen in campaigns against AI — unless you'll go out of your way to do this yourself. Rockets from three boats leave even the flimsy Zeppelin barely alive (assuming units are fresh), so they aren't a threat for fleets, but usually some decent unit like a battle helicopter or destroyer on patrol/flanking pass suddenly gets attacked by 2-3 speedboats and is forced to flee for repairs while other forces drop everything to cover it, because now it's weakened and won't survive if it tried to stay and fight the boats — rest assured, there are more of them around.
Steel Panthers, especially in games set in/after the 1960s, feature omnipresent Soviet ATGM teams: first Malyutkas, then more advanced Fagots and Konkurs. They are size zero (meaning damn near impossible to spot at 1000+ meters), have just enough accuracy to seriously blunt any armored advance, and are deployed in huge groups. Now you know what the Israelis felt in during the Yom Kippur War.
XCOM: Enemy Unknown, late in the game, turns Thin Men into this. They show up en masse during bomb disposal missions, and while you're probably powerful enough at that point to one-shot them, they come in big swarms, like to Reaction Shot your soldiers, and frequently spew poison clouds everywhere, chipping down your soldiers even more.
Drones. They are usually just there for healing the main unit of their group, and when confronted, can only deal minor damage to your soldiers. But if it happens to be the end of a mission and your squad is all on very low health, Drones' ability to fly over all cover can be devastating. Even more so with Red Fog Second Wave option enabled, where the normally harmless hits dramatically lower your aim.
Enemy Within has the Seekers. They don't do a lot of damage with their piddly little plasma guns, but they are supremely irritating because they can turn invisible and strangle your troopers. Unless you drop everything and put your squad on overwatch in a big clump (which then puts you at risk for grenade attacks), they come right the frack out of nowhere and strangle any soldier unlucky enough to be caught alone. If you're trying to go for meld canisters, this is very punishing.
In the Civilization games, barbarian units. They pop up occasionally and will sack and pillage undefended cities and terrain improvements until you defeat them. While they're only rarely a serious threat, they do often force players to spend resources dealing with them that they would rather invest elsewhere.
In Civ V, melee and gunpowder infantry units can count as this, especially once the tech level is up to Great War Infantry or beyond. They tend not to be difficult to defeat, but they are nonetheless annoying if multiple attacks just can't seem to kill them. They can also inexplicably cause damage to units they realistically shouldn't be able to hurt, like planes.
Sid Meierís Alpha Centauri: The equivalent of Barbarians, the frickin' mindworms. Any reasonably faction with even a halfway decent military can usually fight them off, but they are so annoying. Unless you have a high Planet rating—then they're God's gift (you can capture the worms, so they're free units!). Also annoying are Xenofungal blooms, since they come out of nowhere, destroy the improvements you spent so much time building, and mean you'll have to spend anywhere from five to ten turns (or more!) clearing the Fungus and rebuilding what you lost.
Also from Alpha Centauri, Sister Miriam tends to 1) send huge waves of troops at once and 2) equip them with outdated armaments due to her faction's distrust of advanced scientific research. To a player with a competent research program, any one of them will just nick your defenses. On the other hand, 12 of them in one turn?
Genjuu Ryodan has a lot of units that will make clearing maps slower, which is a bad thing as all maps has a turn limit, but once the player gains them these units can be used to return the favour:
The airborne-mermaid-thing can put units into sleep efficiently and two of them using the same sleep attack on the same unit will mostly render the unit fully sleeps, which immobilizes the unit and preventing them from attacking. And the skull-headed version can actually attack units with high attack power on top of the sleep spell being upgraded.
The medusa is like the airborne mermaid, except that it inflicts petrify which is permanent unless being cured.
The genie and the air dragon, despite being an air unit, are extremely tough, deal significant damage to most land units and most anti-air attacks are inefficient against it.
The land dragon is even worse. It is immune to lava damage (which actually gives terrain bonus), can stock up to 16 attacks (most units get 10 at most) and not even the genie and air dragon are efficient against it.
The octopus and giant fish are naval monsters similar to the genie. Both are extremely tough on sea and the former has anti-air attacks. For the latter, thanks to geo effects not even the genie and air dragon above are efficient against it.
The ranger, priestess and samurai has powerful range 3 attacks while other units get attack range of 2 at most, which means they can strike units from afar first. Their primary attacks are anti-air as well.
The phoenix is completely untargetable once they used their only attack, which then generates more mana than the mana generator units, which allows other units stated above to be spammed almost constantly. The only saving grace is that they do not count as units when said attack is used in determining the kill all enemies win condition.