The Mongols are scary as hell in the open field, and are hard to beat back. However, once they hit your cities or a defended bridge, their horse archers are just cannon fodder for your spearmen, archers, and crossbows. The Timurids on the other hand... there are a couple ways to defeat them at their own game in the field, but all require locking down their archers. And cavalry. Lots and lots of cavalry.
The Britannia campaign in the Kingdoms expansion for Medieval II has William Wallace's army, which is presented as an enormous and impressively badass army, with fully-armed and armored and high experience troops. In a straight fight, they'd be tough to beat... except that Wallace himself is an infantry general in an otherwise normal unit of Highland Nobles. That means is that all it takes is one well-timed heavy cavalry charge, and....
Also from Britannia is the arrival of King Haakon IV of Norway and his army. While he definitely brings a formidable force with him, Haakon himself will probably succumb to old age within a few turns.
The entire Aztec Empire can wind up being this if you didn't know you were going to face them. If you put off exploring westward on the ocean (easy to do, especially if you neglect moving up the naval tech tree) by the time you figure it out you'll have conquered most if not all of the available territory and probably eliminated the Mongols and the Timurids. Then you reenact what those two civilizations did and bring in a dozen massive armies into enemy territory and play the part of The Horde all on your own.
Demonic Spiders : Inquisitors, if you tend to be generally irreligious and playing a Catholic faction. What's this awesome general with plenty of good traits but terrible piety? Hope you won't mind seeing him executed for heresy.
Missile cavalry. Especially the ones with armor-piercing and a good defense rating. Jinetes, Dvor Cavalry, Mongol Heavy Archers... pick your poison.
The Russians can gain access to Cossack Musketeers, a top-tier gunpowder unit, the instant blackpowder weapons become available in any city with the Huge Stone Walls upgrade (which is not that hard an upgrade to built to). Watch the Mongols, the most dominant and feared force since the Roman legions and the conquerers of the eastern world, ride up to your lines and die in droves as they are outranged, outgunned and their mobility advantage is rendered useless.
Scotland is a notorious one, and you wouldn't think so as they're barred from using gunpowder units and have relatively weak archers. The reason for that is when you have 1,200 screaming, painted claymore-wielding maniacs, you don't need bullets or arrows, and they certainly won't save your enemy's ass. Highlanders are cheap-as-chips and utterly devastating on the charge, Highland Nobles are also half the price of the elite heavy infantry of other factions and when fully upgraded will carve through infantry and cavalry like a chainsaw through tapioca. Noble Swordsmen are even worse, they're the best heavy infantry in the whole game, no question. Cavalry? Pssh, Scottish Noble Pikemen are unbeatable head-on, can hold a gate against whole armies when fully upgraded, and will even poke elephants to death. The Stainless Steel mod makes the Scots even more broken by inadvertently buffing pikemen through reworking the combat mechanics and letting any castle in the British isles recruit longbowmen with the right archery range upgrades. Pikes + longbows = nearly unbreakable pike-and-shot formations which render just about any assault a sad joke, about three centuries before the development of widespread early firearms resulted in the historical development of this tactic. And what's worse, in the 6.4 version of the mod, Scotland can get William Wallace and his army just like in Britannia: two full stacks of veteran infantry light years ahead of your existing recruitment capabilities each with a brilliant high-level general at their head. The Scots can easily go from struggling for the survival to dominating the entire British isles in a few turns - if you're playing as the English, you will lose your starting army and everything north of Nottingham.
The Moors' Camel Gunners. Excellent mobility and speed comparable to light cavalry, a highly accurate and armour-piercing ranged attack which reloads faster than any other gunpowder unit, they can shoot on the move, and inflict a hefty debuff on enemy cavalry due to freaking out the horses? Yes please.
In the Americas expansion, New Spain starts out with an elite army stack with Hernan Cortez, a ridiculously powerful cavalry bodyguard with eight Command stars and a morale bonus, and culverins, large artillery pieces which can one-shot just about any tower or wall and blow gaping holes in native formations. You can pretty much conquer most of the Aztec empire with just this one army stack.
Game-Breaking Bug: Not exactly game-breaking, but the vanilla game has the two-handed weapon glitch in which the unit constantly blocks the enemy's attack and doesn't even try to retaliate as often as they should be. This means they will be stunlocked until they are poked to death. The Holy Roman Empire, in particular, takes a big hit by this since half of its infantry consists of two-handed sword units.
Goddamned Bats: Quite a bit, which while realistic just slow you down and are generally annoying.
Heretics, especially high-piety ones. Can be useful to train priest's piety (or assassin's skill) if you bother to do so, but they generate religious unrest and WILL make your cities rebel if you don't care about them. Getting rid of the repeated heretic appearances soon becomes a chore.
Witches. They are rare, but they will curse (yes, in a realistic strategy game) your generals with negative traits and be a pain.
Random rebel armies, which at least give you a fight and are generally nonthreatening. But they move around, block your path, and create destruction in cities. Hell, especially peasant rebellions, which are just unsatisfying to defeat.
Nintendo Hard: As a Western European country, conquering and holding the Middle East. Firstly, you have to ferry entire armies across the map, which will likely be most of your military strength early-game. Secondly, all of these regions are Muslim, so you will either need to bring priests, or build the churches and hire more there, but either way, you'll be stuck in a place far from your capital governing people who hate you for being Christian. Third, moving any troops out to go campaigning will likely result in riots in the city they came from, so you'll have to dedicate a full stack just to keeping the peace. Fourth, the Mongols and Timurids both show up in the region with enough troops to conquer every city and castle from Constantinople to Cairo. Unless they choose to attack Russia, your occupation of the Levant is living on borrowed time. With all this considered, it's far easier to save the "Conquer Jerusalem" victory condition for last.
Only Six Faces: Your generals and agents are given a rather small of avatar portraits.
The Scrappy: The Pope. He WILL be a constant thorn in your side if you are Catholic faction and if you wage war on your fellow Catholics. Well, unless you get more votes into the college of cardinals in the midgame, kill the current pope and put up a loyal cardinal of yours as a borderline puppet pope. Or pay him often, but this is far less effective.
Scrappy Mechanic: Unlike Rome, there is no way to change your Faction Heir in this game. Presumably it was done for historical accuracy, with the computer typically selecting the oldest son of your Faction Leader if one is available. (If not, his next oldest brother is typically chosen.) This can lead to scenarios where a greedy/corrupt/disloyal governor of a backwater territory is named heir over his much more qualified brother/cousin who is a loyal conquering general. Your only option at that point is to send your undesired heir off to battle as part of a Uriah Gambit (and hope against it turning into a Springtime for Hitler situation where surviving makes him even harder to kill in the future.)
Those Two Guys: The knights who always chase down your assassin after he fails.
Ugly Cute: Princesses can have high charm even when they gain ugliness traits. It is indeed quite possible to have a maximum charm princess who any foreign prince would love to marry, but whose face has earned her an unfortunate name like "Agnes the Gorgon".
What Measure Is a Non-Badass?: Sparing captured soldiers gets you chivalry points. You can then order a massacre in a city (which can give you dread), but doing so reduces squalor, increases profit, and lowers disorder. This trope comes into play because town will then get tax and population growth bonuses from being run by a "chivalrous" general. That's right, you can spare a few dozen soldiers and slaughter thousands of civilians and still get called "chivalrous" by the peasants.