Labyrinth Tanks. If they're the leader of a team, they move incredibly fast, and once engaged in combat they hit very hard and are pretty tough to take down. It usually takes a few rounds of combat in order to take down a team with a Labyrinth Tank.
Fusion teams. Unless you can knock out one of the fusion components before they use Polymerization, you will almost definitely lose the fight or be forced to run.
Disc-One Nuke: In the Badlands mission, you can encounter a team led by The Illusory Gentleman. If you defeat them, then they will join your army. Unlike almost every other monster, the Illusory Gentleman will join your army at Level 30, complete with boosted stats and AP. He will easily be the strongest monster in your army for the next few missions, and is a reliable candidate for even endgame-level missions thanks to his enormous head start.
Ansatsu is given to you for free when Bakura joins early in Yugi's story. It has amazing stat growths, lots of AP, and a surprising amount of health. It's one of the all-around best monsters in the game, and considering how early it shows up, it could also be called a Disc-One Nuke.
Use Metalmorph on Red Eyes-Black Dragon and you will get Red-Eyes Black Metal Dragon, arguably the best all-around monster in the game. Its stats grow at an insane rate and it gains AP quickly. After a bit of training, you will have a flying tank that tears off massive amount of enemy health with each attack and is absolutely swimming in AP.
Give Mai all three Harpie Ladies, and equip each one with a Cyber Shield and/or an Electro-Whip. She now has a trio of monsters with tons of AP and a very high chance of paralyzing opponents, as well as a powerful special attack that hits all enemies. The Harpies are easily one of the best teams in the game in almost all situations, but especially when it comes to neutralizing enemies with items.
Goddamned Boss: Bandit Keith's second fight if you're trying to get the rare monster. He's not too difficult, but Keith constantly charges you, meaning you have to flee from him repeatedly as beating Keith immediately ends the mission. Making it worse is that you have to do this twice if you want all of the monsters.
Player Punch: Many of the special Game Overs make you feel awful for losing, such as having Yami Bakura burn down Fizdis's hometown while her parents are in the city, and Scott forcing Joey to sacrifice himself if you defeat him without Mai.
Scrappy Mechanic: Wandering monsters. While some locations are straightforward, most monsters are hidden in places on the map that are out of the player's way, and are nearly impossible to find without a guide. Some are even more difficult, requiring a specific marshal or a specific monster in the party that encounters them. And then there's Moisture Creature, who is a moving roaming monster.
Arkana, especially if you did Yugi's campaign first. His Dark Magician is Level 99, and if you fight him as Yugi—which he tries to make you do—he'll show off his Raigeki spells, which deal heavy damage to each of your monsters. To make matters worse, if Yugi loses you fail the entire mission. You could be seconds away from victory only to have to do it all over again.
Scott Irvine's robot stand-in in Kaiba's campaign, especially if you already completed Yugi's campaign. You can only bring three marshals, and he rushes you with seven teams. Scott's own team is no pushover, as two of his monsters have 7 Blue Medicines and 3 Red Medicines, and all of them have a fair amount of Action Points and HP.
That One Level: The second fight with Yami Bakura gives you about 30 seconds to reach Jakhud before he does. Fail, and it's an instant Game Over.