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.
Goddamned Bats : Quite a bit, which while realistic just slow you down and are generally annoying.
Heretics, especially high-piety ones. Can be useless 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.
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.
Those Two Guys: The knights who always chase down your assassin after he fails.
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.