Enter the Matrix has three driving levels. If you play as Niobe, you get to be the driver while Ghost takes shots at the enemy vehicles, and if you're playing as Ghost you get to be the gunner while Niobe drives through the level. The problem? Apparently the AI-controlled Niobe completely flunked out of driving school, because she can't go five seconds without crashing into something and more than likely getting stuck (this is most aggravating in the final driving level, where you're trying to escape from the Twins, who are following after and shooting at you, and are also completely invincible.)
Space Pirates, in the stealth section of Metroid Zero Mission, will raise an alarm and mercilessly chase you if they spot you. However, you can cause them to call off the alarm if you can keep them from spotting you for a short period of time (or going to a prescripted area to shake the heat). This is despite the fact that you are the one solely responsible for the destruction of their leader not three hours ago and you are now unarmored and vulnerable. It's also worth noting that the shots they fire at you will kill each other if you can line them up right.
Similarly, Serris-X, and B.O.X have blind spots where they will be completely unable to hit you (Serris-X) or just stop doing anything to try and hurt you (B.O.X). B.O.X II has a similar blind spot, but it's not as foolproof since it'll still shoot at you with (easily destroyed) missiles.
In Castlevania: Dawn of Sorrow, you eventually encounter a human boss who can use any power used against him, but it is automatically overridden by any new power. That's fine, and obviously the best way to beat him is to use a stupid power against him. The trouble is that he never clues into the fact that his new power is ridiculous (he does, after all, have a knife he could be using), such as lashing out with a Cave Troll's tongue attack that doesn't extend past his dramatically outstretched arm.
Sadly, his Knife is even MORE puny then his new Tongue attack. His power could be a Game Breaker anyway if he had any actual control over it and if the powers he copies weren't limited to lv1 out of (generally) 9, which the Cave Troll soul is the most extreme example of.
Use the Student Witch attack on him, so that he spends the remainder of the battle trying to throw cats at you.
This is actually a bit of a Guide Dang It, since the game never tells you about the trick outright (sure, he visibly steals his primary attack from a Malachi, but there's no indication it's automatic) - and if you don't know it, the fight is tough. (It's also tough in Julius Mode - turns out that power works on subweapons, too. Sure, you can give him Yoko's "power palm", but if you forget about it and hit him with an axe?)
The enemy AI in the Armored Core games, especially on the PS1 and PS2, are capable of truly staggering feats of incompetence. Choose to fight AC's in the right arena and they will:
A) Attempt to get at you by futilely trying to phase through solid matter.
They have also been witnessed boosting out of the combat area for no reason, giving the player the victory by default. As you can imagine, there are myriad ways of rapidly climbing the arena ranks by exploiting the stupidity of its inhabitants. But the real problems start when From Software, rather than attempting to program better AI, decided to compensate for the computers' stupidity by giving the AI controlled ACs capabilities that far exceed what is possible, or sane, and in Armored Core 2 even equipped the AI with parts that didn't exist.
In the stealth sections of Batman: Arkham Asylum, the Mooks rarely ever bother to look up. It's a little bit more frequent in the harder difficulty levels, but still.
They also have no periphral vision whatsoever, except for the insane inmates who have a perfect line of sight.
If it wasn't for their constant conversations, you'd assume they were all deaf, too, given you can stick one of them in a Dragon Sleeper (complete with barely muffled groans) with their buddy none the wiser ten feet away.
Those insane inmates actually have a perfect line of sight for Batman to the point of this trope. It's uncommon, but it's very possible for those inmates to run into obstacles, each other, or to trip on scenery trying to get at the player, causing them to knock themselves or other insane inmates down for an easy knock-out.
The guards in Assassin's Creed I will sometimes throw you off a high ledge, then jump down after you. You can survive the resulting falling damage. They can't. In 2 you often lose thieves to the idiots trying to keep up with Ezio and jumping from too high or failing jumps.
The Multiplayer Tutorial AI dummy in Assassin's Creed: Brotherhood won't care if the player approaches him in an unusual manner, and will only jog away from the player if a chase is activated, never sprinting.
In 2, Brotherhood, and Revelations, Ezio can swim. No one else can. This does not prevent anyone moving near the water trying. It can be very frustrating when one of the super-elite assassins you've spent hours training makes a perfect kill on a guard in the docks, then calmly steps off the pier to his death.
In Metal Gear Solid 2: Sons of Liberty, the guards provide a fairly solid challenge without going to brutal measures to catch the player (difficulty dependant of course). However, patrolling guards, when not faced with a left turn, will ALWAYS turn to the right including when they are simply turning around. This effectively means that the player can stand next to a patrolling guard and not be seen, providing he always stands on the guard's left side.
Which is far from the worst or only problem. For instance, you can shoot guards with tranquilizer guns, which mostly avert Instant Sedation except on Very Easy difficulty, but regardless of how long it takes, will cause the target to abruptly keel over with a grunt and start snoring. Other guards will find nothing unusual about this if they find a sleeping guard, and will simply kick them awake, even if they saw them keel over. This is particularly silly when you hit a guard who regularly sends status reports by radio (or interrupt a guard with a radio), and a group of armed soldiers come to investigate. After kicking him awake, one of the soldiers will radio back to report that there was nothing wrong before they leave.
Speaking of Metal Gear, Metal Gear suffers from this in both the MSX and NES versions. Sure, one can't expect a complicated AI all the way back then, but so long as you're not directly in their literal line of sight, enemies can't see you - even if you're standing just to the side, where you can literally run right past them or even attack them repeatedly without them reacting in any way, so long as its a silenced firearm or with your fists. Not even other guards dying in their sight line makes them react. And when you do alert them, they just run in straight, grid-like lines spamming bullets infront of them with no real thought other than reaching you.
Metal Gear: Ghost Babel suffers from similar AI, practically being a Spiritual Successor of the MSX games with elements from the first Solid title. The big difference is that enemies can face diagonally now, which means they can turn to see you at least, but all the problems from the original games still exist in the form of very narrow line of sight, probably literal on 'line', in whichever direction they face. Thus you can still literally run past them and brush shoulders even, so long as Snake's hitbox doesn't touch theirs or their invisible line.
In Dead Rising, it's not uncommon for Frank to be escorting a couple of survivors and, even though you've given weapons to as many of them as you can, for them to stand there calling for help while they're being eaten alive by zombies and doing absolutely nothing to defend themselves. This can be especially frustrating if you're handling a survivor that can't carry a weapon or if you yourself are in the middle of being attacked. This is even MORE frustrating if you were attacked while trying to help the idiot and you all die because said idiot will not even push the zombies (all the survivors are capable of pushing).
They also have no concept of retreat, and will stand there fighting off a horde of zombies, no matter how overwhelmed they get. Leading to the tactic of mashing the call button to make them move their sorry asses.
Also, don't give them a gun. Unless you like getting caught in friendly crossfire.
In Gears of War, Locusts (the main enemy in the game) are supposed to dynamically move around and take cover in response to your team's position. However, nine times out of ten, they will, in a pitched firefight, leap over the cover to reach a better place, leaving them horribly open for an explosive headshot.
In the sequel this was fixed, but the AI has even more pitiful failings; enemies will run straight into security lasers, clearly-visible proxy mines, a sentry turret's line of sight, etc.
When given an explosive weapon AIs will choose to destroy themselves. But only if you are not within range.
In Grand Theft Auto III, random emergency vehicles will sometimes speed up the drive to the mafia don's house, slam headfirst into his garage door and continue to grind against it until their vehicles explode.
Everyone in Liberty City (apart from Claude) seems absolutely incapable of aiming a rocket launcher in any direction but down. And they actually seem to be aware of this, since if a pedestrian were to have a rocket launcher on them, they would run up to their target, and fire the rocket at the ground, killing both the target and themselves in the process. By San Andreas, this has been corrected so that pedestrians can fire rocket launchers at what is in front of them. *gulp*
In Transformers: Revenge of the Fallen: The Game, the enemies often have Ungodly Dodge abilities. However, this is often counteracted by their tendency to stand near gas tanks and then blow them up.
And that's not the end of it: in the rare occasion that you have backup with you, it will blindly charge into battle and be destroyed, thus leaving you to complete the mission by yourself, one hundred times more efficiently than if they were around to help.
Resident Evil 5 AI varies. Sometimes it's great, picking off enemies with relative efficiency with reasonable choices in weapons. Other times, if you're laying claymore mines to set up a dastardly trap while fighting a big boss, the AI will quietly follow in your footsteps picking them up.
Sheva's AI reaches the pinnacle of stupidity in the first battle with Wesker. First she stands still and gets badly injured (or killed on the higher difficulties), then runs off to hide, then TRIES TO FIGHT Jill ALONE and, as a result, dies. If she's lucky enough to survive, then she'll try to link up with you, leaving her side of the fight unfinished and bringing a very dangerous combatant with her to help Wesker. You cannot complete this fight solo without hiding.
It's pretty much concluded that this was a lot nastier than just bad programming, that they intentionally made Sheva's AI unbelievably stupid to make the game practically unwinnable without a second player, which often requires a second copy of the game, second system, and online accounts.
However, if you give Sheva a Sniper Rifle and a handgun and keep her relatively stocked with ammo, she manages to become relatively useful.
The AI in Resident Evil 5 does not understand how explosives work. At all. No matter how many you give it, no matter what the situation, no matter if you take all of its other weapons away, your AI partner will never, ever use grenades, grenade launchers, or mines. The game never tells you this, and your partner will still happily clog their inventory with items they won't use, though.
In the very beginning of its predecessor, Resident Evil 4, there is a section of tripwire that will detonate upon contact with either the player or enemies. The pissed-off villagers apparently forgot who set the explosives in the first place, because they will run right into it and kill themselves if you position yourself right.
The Mummy Returns is hardly a pinnacle of gaming history, and the AI Medjai shooting at you will often miss at super-close range if you run back and forth a lot, but the best is the boss fight against Ardeth Bay for Imhotep in Cairo, the second mission. After killing all of his Medjai guards, you can simply back him into a corner and mash the kick button, resulting in a never-ending stream of kicks to the head that Ardeth can never get past or block. Since you can't kill him and just have to hold him back until your train leaves, this makes the boss fight less of an issue than the health-draining cats.
Driver 2 Advanced really did stretch the bar at the time for the Game Boy Advanced. It was remarkably fluid despite the pixel count being lower than fifteen for particular sprites. The control scheme and driving performed decently for the extremely limited physics engine it was rooted to. But the catch here is, a felony can be really unpredictable unless you try and figure out how to piss off the police and blinking pedestrian sprites. Why is this stupid on the A.I.'s end? Particular illegal crimes such as driving on the sidewalk and stealing cars would occasionally become "legal" and no authorities will pursue.
In The Godfather, expect that pedestrians will somehow, in an attempt to jump out of the way of your car, instead fling themselves into your path. Especially annoying, when you consider that any contact between a moving car and a civilian, at any speed, is almost universally a One-Hit Kill.
In The Uncanny X-Men for the NES, one-player mode would saddle the player with a computer-controlled ally so unfathomably stupid that players found it most convenient to get the enemies to put it out of its misery.
In Grand Theft Auto IV, enemies will attempt to use the cover system just as much as you do. While most of the time, enemies will use this cover effectively, they do on occasion take cover behind objects that are not very good at deflecting bullets or won't cover them effectively. So, you may spy a supposedly highly-trained N.O.O.S.E agent take cover behind a stack of cardboard boxes or a fire hydrant.
The AI used in the street races drives like a blind moron on drugs. It usually ends up overshooting corners because it doesn't slow down in time, plows right into other cars instead of trying to dodge and occasionally even manages to veer off a straight road.
Fly up in a helicopter with a high wanted level. Hover over a body of water (the ocean next to Francis International is a good example), and watch as the police attempt to pursue you. They seem to not understand the concept that they can not drive their car up to you, or that there is an ocean in the way. After a few minutes, half the local precinct will probably be bobbing about in the water like corks.
Ambulance drivers are completely out of their minds in GTA IV. They fly around at top speeds, overshoot turns, veer off at the slightest provocation, and crash into everything in their way. There are reports of ambulances mowing down five people in an attempt to resuscitate a guy who punched in the face a few times.
In the first Tenchu game, you can swim but your enemies have Super Drowning Skills. They apparently don't know this, because if you are spotted by an enemy you can simply jump into water and watch them follow you in to drown.
Sometimes, the AI for The Force Unleashed can be do something brilliant and organic, but more often it tends to be dumb as rocks. It isn't unknown or impossible to literally sit behind a force field wall doing nothing and watch a bunch of stormtroopers and a purge trooper effectively trip over one another and fall off bridges, ledges, and cliffs. It's possible to see almost an entire squad kill themselves with no input on the part of Starkiller.