Basically, this is an enemy who makes you chase them around or stake them out for ages in order to defeat them. There are three main forms:
An enemy who turns tail and runs away from you as soon as you spot them, forcing you to give chase. They may occasionally stop briefly and fire back at you, before resuming running. If you have a long-range weapon, they will probably zigzag about, making it hard for you to aim. If you are in a closed arena as opposed to a one-way tunnel or path, the enemy may run in random directions rather than just forwards, allowing them to catch you should you fall too far behind or take a wrong turn.
An enemy who teases you, coming out to spray a load of bullets, before jumping out of the way, diving down a hole, or running off as soon as you get close enough to attack. When you fall back a bit, they'll reappear. They are, needless to say, difficult to hit.
An enemy who is stationary in a small lava pit or pool of water, etc. Like the previous sort, they will come out to attack, but once they have finished firing at you, or as soon as you take the offensive, they will sink down into their pond, where you can't get them.
So why bother with them? If they want to run away or hide, why not let them? If only it was that simple. Alas, you have to kill these things if you want to progress. Maybe they have something you really need; maybe they are blocking the door or path ahead; maybe they cause so much trouble when they do come out that you can't concentrate on anything except dodging their crazy gunfire, and would quite like them out of the way so you can get on with whatever it was you were trying to do. Most likely, though, is that they are a Boss or mini-boss who you have to get rid of in order to continue.
Sometimes they can be fairly easy to catch if you time your move right, but often they fall into Demonic Spider territory, taking multiple hits despite the fact that it takes ages of sprinting, ducking about, or careful stalking just to get them once. If they are a common occurrence throughout a game, they can easily become classed as Goddamned Bats.
Occasionally, they are a not an enemy at all, but an item, creature, or person you have to catch, but not harm, in order to use, ride, or obtain information from.
The difference between this and Cowardly Boss is that a Cowardly Boss will run, but occasionally stop to give you a chance. Get Back Here Bosses won't give you a chance at all if they can help it, and will only stop or emerge when they're attacking you. They won't pause to let you catch up, or stay up long enough to let you aim before firing.
If you're chasing the boss in a circle, it's Chasing Your Tail. Not to be confused with You Will Not Evade Me (which is when the boss decides that it is not going to allow you to pull these sorts of shenanigans on it.)
You may have this start playing in your head while fighting some of these.
Majora's Mask has Wizzrobe, a normal enemy in other games but a miniboss here, who also does the whack-a-mole method, as it fires, disappears, and appears elsewhere to fire again. (They do that in all games, but in this one, there are several specific points where he can appear. You have to spot him, avoid his blast, and hit him before he can disappear again. Like any good Zelda villain, hit him enough times and he Turns Red before you can actually kill him.)
Not to mention Majora itself, who in the second phase of the boss fight spends more time dancing away from you than fighting.
Goht is a mechanical bull who runs in a circular track and fires back at you, never stopping. Then he resorts to rocks, and bombs. The way to defeat him is chasing him as Goron Link and bumping into his legs to make him trip. (You could also shoot Fire Arrows at his forehead, but the former method is more fun.)
And of course, there's Stallord from The Legend of Zelda: Twilight Princess, who, after seemingly defeating it in a normal type battle, has its head come alive and fly around the area shooting fireballs, requiring you to chase after it with the Spinner.
Then there's the Giant Spider boss in Twilight Princess, whose Clipped Wing Angel form will run randomly around the boss room faster than Link. Easiest way to finish it off? Stick a bomb arrow in its ass.
Link's Awakening has a couple. There's the miniboss Master Stalfos in Catfish's Maw, whom you have to chase to four separate rooms before finally defeating him; the Slime Eel, the main boss of the same dungeon, does the whack-a-mole approach where it occasionally pops its head out of holes in the walls and has to be pulled with the hookshot to expose its weakpoint. Also, the Grim Creeper in Eagles' Tower; after you defeat his bats, he runs to the top of the tower, where you have to fight him (on an eagle) as the main boss.
The Legend of Zelda: Spirit Tracks has Byrne, who constantly jumps back and forth between a series of posts around the arena for much of his first phase, all the while peppering you with weak but quick energy blasts. The only way to get him down to your level is to wait until he launches his grappling claw, dodge it, and direct Phantom!Zelda to grab it and pull him off his post.
The Legend of Zelda: Skyward Sword has the Imprisoned, who completely ignores Link as it marches toward the Sealed Temple. Link has to hit the Sealing Spike on its head before it reaches its goal.
The infamous Water Dragons from Castlevania III: Dracula's Curse, who emerge from instant-deathwater to immediately breathe fire that knocks you off of one of three small platforms to your doom, and then dive back in without giving you any fair chance to hit them once you're through cowering from their attack. What makes it worse is that they have almost no pattern whatsoever, meaning you can't predict where they'll appear of which direction they'll be facing, so you can't try and outsmart them and attack from behind. Your only chance is to walk to the opposite edge of the center platform, hoping that the dragon head will either be high enough to hit with the Axe or that your whip will be timed just right as it drops back down to make contact with it. And you have to be as careful as possible because, thanks to the aforementioned Malevolent Architecture if you screw up even once you have to do this ALL. THE FUCK. OVER AGAIN.
The Shrewd Possessor from Luigi's Mansion: Dark Moon is like this, and the Boss Battle is very unlike any other in the game (it's sort of like a cross between a bobsled and a shooting gallery). After it takes the form of what resembles a large, monstrous face made of ice (more or less) Luigi has to chase it down a mine shaft in a motorized sled and lob bombs at it to break off its armor, at which point it makes a single attack. If Luigi manages to hit it with one more bomb in its mouth when it tries, the chase ends for the moment, and Luigi is able to use the Poltergust 5000 on the Shrewd Possessor. (The whole process has to be done three times to defeat the creature.) The Health Bar is still in use here, but the greater danger is the sled's engine overheating (which happens if the gauge goes too high, and taking too long or failing to dodge that one attack causes it to go higher).
Lego Batman. Harley Quinn spends the whole fight backflipping away from you in a circle.
This is the behavior of JawsBruce the great white shark in LJN's Jaws for the NES. When you encounter the shark, he will roam lazily around in the regular shooting stage screen that ensues, and once the regular enemies have all come and gone, if you haven't depleted his life meter, you go back to the ocean map, and he recovers some of his hit points.
Sanctus from Devil May Cry 4 fits this trope. To make it worse, as well as putting up with his endless taunting while he plays keep-away, you have to attack him to break his shield (dodging all kinds of projectile fire all the way), and if you fail to do this perfectly, he attacks you for massive damage, then regenerates his shields, forcing you to begin the sequence again. Even worse than that, you have to fight him twice throughout the course of the game.
Denkou from Astro Boy: Omega Factor does nothing but run away and leave behind bombs with increasingly short fuses. For added fun, she's invisible, so the player must deactivate devices before chasing is even possible.
Wangan Midnight Maximum Tune 2 and its successors have the "Feeling Team" stage, where you start 3.5 kilometers behind and have to catch up before the team reaches the goal. Fortunately, the Comeback Mechanic and the team having a slow car make it possible.
Any AI opponent fought in the Planet's Core or Ultimecia's Castle stages in the Dissidia: Final Fantasy games. Both of these stages are vertically oriented and have spiraling elements going from near-top to near-bottom. And the computer, for some reason, just loves to grind along the spiraling elements, up and down, all day long. Their is no benefit to them from doing it, and while grinding they are basically the fastest thing on the stage — leading to an extremely frustrated player dashing at them again and again and again, trying to catch them so they can fight already.
First Person Shooter
Descent and Descent II have a boss that unloads a payload of high-damage weapons, and teleports away. To a lesser extent, Descent II and 3 also have a thief bot that tries grabbing equipment, and running across the map if detected or attacked.
And also Margaret, who is type one. Luckily, she's not that hard to beat, and even engages in melee when Travis is close enough.
The battle with Hermes in God of War 3 is spent mostly just trying to catch up to him, all the while he taunts and mocks you for being slow. When you finally catch him, he doesn't put up much of a fight, but on the plus side, you make him eat those words...
Sengoku Basara has several examples. There's Hideaki, who just won't stop running away and who most objectives involve pursuing; Tsuruhime, who "jumps" all the way back to the beginning of the stage (luckily, this can be avoided if you kill all the soldiers quickly enough); and Yoshihiro, who disappears before you can defeat him and will only come out once you've taken all his bases.
Haruhisa is the most pronounced example: he not only runs, he hides under the dunes of his desert level, with a sandstorm obscuring his whereabouts on the minimap unless you capture a base he happens to be close to. If you find him, you get to fight him for a minute or two, after which he'll call up a sandstorm again and retreat to an area still under his control. If you capture all of his bases without defeating him, he retreats to a cave in the middle of the stage.
All enemies marked as "Ranger" in Divinity 2 will always flee when in range of your Melee, which makes them rather Demonic Spiders when they can stun and poison you while fleeing, although that's only when they are surrounded by other allies who are more than willing to get in your face with stronger attacks and block your pursuit.
In Lost Souls MUD, the most notorious offender on this count is Finwe, who, if seriously wounded, is perfectly willing and able to teleport himself to an extraplanar refuge and heal up before continuing battle.
In City of Heroes, some missions require you to arrest all enemies on the map. This is a problem when facing the Casters for the Circle of Thorns, who will run after you give them a good whack, requiring you to chase them through what is basically a maze, and you will probably run into a SECOND group of enemies while chasing the one survivor from the first. This is even more irritating when facing the Cabal, who do the same thing but can fly, allowing them to run to high places very quickly. Attempting to ignore these enemies only means they will find you later on, usually as you're down to half health after moving on to the next mob. Finally, in PvP battles, players who can fly do this to those who can't, allowing them to heal and sneer at the grounded player. This is where the power Gravity Control comes in handy...
The Teleportation power pool. Nothing to prevent it (except stunning) and it gets very annoying when your enemy vanishes in a flash. In PvP, you can at least predict where the foe's gone, because PC teleportation is limited by line of sight. Then, you meet those blasted Sky Raider Porters, NPCs who can teleport through walls - preferably at the other end of the area.
World of Warcraft has a few of these, pretty much any of the multi-stage bosses. A good example would be Razorscale from the Siege of Ulduar; before you can even fight her, you have to construct turrets to pull her down to the ground, and after a few seconds, she'll break the chains and fly again, forcing you to start all over.
There's a variation of this in the much-hated Halls of Reflection dungeon; you are the Get Back Here Boss. The final part of the instance consists of you frantically trying to run away from the Lich King as he chases you... slowly... throwing up ice walls in your way as you demolish hordes of his mooks.
The giant mole in RuneScape. For an added bonus, it will put out your light source if it is an open flame.
Nearly every raid boss in EverQuest (and many non-raid monsters as well) have the ability to summon whoever is on the top of their hate list to ensure that the player can't run away. It doesn't matter how far away they are in the zone either.
Ghostring in Ragnarok Online will teleport when his health reaches a low enough level, then continues to teleport whenever he spots a player. He's also invisible 90% of the time.
Inhert in Zone of the Enders: 2nd Runner spends most of the first half of its boss fight flying away while shooting at you.
Eggman does it in Labyrinth Zone from the first Sonic the Hedgehog; you're lucky to get more than 2 shots at him (1 before the obstacle course & 1 after) before he buggers off. However, this time, you win just by surviving his obstacle course.
The 8-bit version has the Bridge Zone boss, which pops out of water (your only chance to hit him), but he fires bullets at you. You could get him a few times before he goes under again, though.
The bosses of Hill Top and Oil Ocean in Sonic the Hedgehog 2 both use the third variation, popping from their hiding places long enough to attack before retreating back down.
Sonic the Hedgehog CD crosses this with a Ring Out Boss (of a sort); in Quartz Quadrant, Robotnik stands on a stationary platform, while Sonic has to keep running on a treadmill moving away from the platform. More details on the Ring Out Boss page. Later, it plays with the trope again as Sonic famously races against Metal Sonic through Stardust Speedway. Key word being "race".
In Sonic and Knuckles's Mushroom Hill Zone, Eggman simply flew away from you, and you had to jump through spiked ball walls and get close enough to hit him.
Later, the game inverts the trope. In Lava Reef Zone, Robotnik will use a machine that pops out of lava and shoots spiked balls at you behind you. You can't harm him directly; you have to wait for the spike balls to hit him by jumping up the platforms. In other words, you win by running away from him.
The trope gets played straight again in the last form of the Final Boss, in which Eggman's trying to flee from you as the Death Egg collapses behind you. The trick is to be able to hit him without getting bounced back enough to lose all your momentum and fall down. From a storyline perspective, this is also what's going on during the True Final Boss fight, with Eggman utilizing a missile-and-laser-spamming spaceship/mecha in a last-ditch effort to shake off Super Sonic's pursuit and retain possession of the Master Emerald.
It's not just the 16-bit outings; Sonic games are full of this sort of boss. The final Sonic vs. Shadow fight in Sonic Adventure 2 (where you must keep up with your opponent as pieces of the ground fall to the Earth below), every single boss fight bar the final boss in Sonic Advance 2, and the fights against Eggman's robots in Sonic Heroes and Sonic Unleashed count too.
Sonic the Hedgehog 4: Episode 1's Mad Gear Zone boss starts out copying the boss from Sonic the Hedgehog 2's Metropolis Zone (which the Zone itself is modelled after), then does this after you hit him enough times. The first stage of Lost Labyrinth Zone's boss also does this, mimicking the boss from the original game's Labyrinth Zone. However, unlike that one, the Lost Labyrinth zone boss actually stops and fights you after the chase.
Many Sonic games nowadays have this kind of battle as a standard boss, as it is very suited to the "Modern Sonic" gameplay style: three of the four bosses in Sonic and the Secret Rings, half the bosses in Sonic Unleashed, and a third of the bosses plus the final boss in Sonic Colors (one pictured) are of the "running battle" variety, where you have to catch up to the boss to attack while also avoiding its counterattacks.
In Sonic Generations, all three rival bosses (Metal Sonic, Shadow, and Silver), as well as the Modern-era bosses of both versionsnote Egg Dragoon in HD, Egg Emperor on the 3DS, all work this way.
More specifically, the Rival bosses are all races on the 3DS version. The HD versions are actual battles, and the Shadow battle... um... can we just call it awesome?
The final boss of Sonic Lost World, while the rest of the game averts this trope completely, also utilises the "running battle" format in a style very similar to that of the final boss of Sonic Colors.
In the Lethal Highway stage of Shadow the Hedgehog, the Hero Mission requires Shadow to destroy the enemy spacecraft, which flees from him continually, and is hard to hurt. This is one case of this Trope where it might truly escape, and if that happens, your only option (other than restarting the game) is to switch to the Dark Mission.
Iron Jungle is the same thing, but this time with Eggman. And again, it can and will escape if you don't take it out in time.
Wait, let's not forget Air Fleet! This time, it's the Dark Mission, but again, it can get away.
The fight against Gnasty Gnorc in the original Spyro the Dragon had you charging after him, hoping that when he paused to taunt you, you were close enough to scorch him once before resuming the chase.
Spyro the Dragon is also rife with Get Back Here Enemies, if you're going for 100% completion or if you're in the secret bonus level. Even worse, they mercilessly taunt you.
Not a boss or even an enemy, but the bunny in Super Mario 64 definitely counts... and you have to catch him. Twice, if memory serves. And there are more of them in the DS remake.
And never, EVER grab the monkey, unless you feel like chasing him around to get your hat back.
One of the stars on Tall Tall Mountain requires you to catch him, though, and like the bunny, he jumps away as soon as you get close. Unlike the bunny, however, he's slightly easier to catch, as he tends to jump over your head instead of just dodging sideways. This allows you to predict his landing and grab him with a quick power dive.
Jackle in the first NiGHTS game pulls this, flying backwards and throwing cards at you.
Mzrmr from Eternal Daughter is of the third type. At the beginning of the battle, he rises up off the screen, and will only descend in order to revive the two flunky bosses after you've killed them, then immediately ascend again.
The dragon Tor Chi behaves this way in Legend of Kay. Thankfully, you don't have to kill him. You "only" have to survive his fire long enough to convince him that you're not a rat.
Mega Man & Bass has an incredibly frustrating battle against King Plane, a heavily-armed battle jet flying backward down an endless corridor while all you have to fight it on are rocket boards flying over a bottomless pit. To make matters worse, you have to keep jumping up and down on the platforms from different heights. Oh... and King Plane can smash them with its fists- and flash bombs that blind you so you can't even see where you're leaping! Worst of all: it's got a photon laser you have to shoot and break, or it will rape your health with one hit! It's bad enough as Bass, but a NIGHTMARE as Mega Man.
Ratchet & Clank: Up Your Arsenal: Inverted during the Tyhrranoid Mother boss battle; you have to constantly run away from it to even avoid its attacks, and the second phase of the battle is literally just you running away from it.
BIT.TRIP has the final boss of RUNNER. You have a final platform challenge while advancing your way until you FINALLY get to Goomba Stomp the boss.
The Seer Council in Dawn of War II: Retribution is this combined with Flunky Boss. Your goal is to kill all members of the Council, who themselves just stand there, but they have a good swarm of mooks to keep you busy and a Warp Spider Exarch who starts teleporting them away as soon as your forces get close. Killing the Exarch is the only way to pin them down long enough to go for the kill. Which is revealed as a Nice Job Breaking It, Hero moment in the next mission.
In NetHack, "covetous" enemies can teleport to you or away from you at will, even on levels where teleporting is otherwise disabled. When wounded, they teleport to the up stairs, and then climb up a level when you approach. In theory, they could repeat this cycle all the way to the top of the dungeon. The trick is to lure them away from the up stairs, and then stand on them yourself; the monster won't think to flee to the down stairs instead.
The third Giant Bowser battle in Mario & Luigi: Bowser's Inside Story is against the Fawful Express, a train that is constantly moving away from Bowser. It stops occasionally to refuel, which is when you can damage it. If you can't defeat it before it reaches a bridge, you automatically lose and have to start the battle over.
The fourth Giant Luigi battle in Mario & Luigi: Dream Team is against the Zeekeeper, who constantly flees from Luigi into dimensional rifts and stuff, forcing Luigi to have to counter his attacks to get a chance to fight back.
Doom Gaze/Deathgaze from Final Fantasy VI does this. If you don't deal enough damage to kill him the first time you fight him, he'll eventually escape from battle and you'll have to find him again. Luckily, he keeps all the damage from previous fights. But don't even think about using Vanish/X-Zone on him, either — if you do, the script which gives you the Magicite doesn't activate.
Like the above example in Lightning Returns: Final Fantasy XIII there's Aeronite, who you have to stagger him in 3 minutes or he decides that you're taking too long and flies away. You have to stagger him 4 times and then take off his massive 11-38 million HP (depending on difficulty) in 3 minutes or he flies away then too. And unlike the above example, he fully heals each time he flies away. Needless to say abuse of the game's bullet time is required if you want to beat him in time.
On that note, cavalry and horse archers in Mount & Blade will not stand and fight. Lancers charge into battle, then charge away, then repeat, while horse archers try to play helicopter as long as their arrows last.
Fighting any Cavalry Archers very quickly turns into one of these in Mount & Blade if you don't have a ranged weapon or aren't properly skilled in horseback archery. If you're on foot, you can't really catch up with them and they won't get close enough for you to hit them beyond abusing the "invisible wall of doom". If you're on horseback, they're going to just keep running away from you taking potshots until one of you dies. And their horses are almost always either faster or more manueverable than yours.
The hyenas in the sequel are probably a better example, since, in order to get information out of them, you have to chase all three down and beat them up. And, if nothing else, they're damn fast (yet hella fun to chase after since there's no time limit).
Xigbar, whose annoying tendencies were already present with his human form, Braig, in Birth By Sleep
In 358/2 Days, the entire first half of Agrabah's boss involves trying to catch up to it while sand is slowing down your running speed. Once it reaches its last life bar things pick up, however.
Ruler of the Sky and Emerald Serenade are notoriously hard to catch up with.
The first game, Chain of Memories, and coded all have Iago. You have to beat him to get Jafar's lamp, while Jafar himself is pelting you with various attacks. They also have platforms shifting in height, often keeping him out of range of your Keyblade (though your spells can usually hit him)
The Char Clawbster, Holey Moley, and Spellican in Kingdom Hearts 3D. The first likes to retreat to the opposite end of its very long battlefield periodically, or worse, to the ceiling on the opposite side of the battlefield. The second pops out of random parts of an enormous room (Including the walls and ceiling) and has a very bad habit of retreating the moment you get in range to attack it. The third abuses Teleport Spam and gives you very little time to attack before it moves (Though you can buy more time by using Zero Gravity on it).
The roaming legendaries of Pokémon, first introduced in Pokémon Gold and Silver and Crystal, where there were three. Doesn't sound that bad right? The catch is, they run away on their first move, even if you put them to sleep (as demonstrated here). They kept this idea in all main Pokémon games after that. It isn't known why these incredibly powerful Pokémon are such cowards, although with one of them in Diamond/Pearl/Platinum (Mesprit), Professor Rowan suggests it simply "wants to play with you."
You think you'll use a Pokémon that knows a trapping move? They're faster than you are, so they move first and can leg it before your move even happens. You've boosted your Pokémon's speed to ridiculous levels and use the trapping move? They all know Roar, which makes YOUR Pokémon run for it - and if you're playing the FireRed/LeafGreen remakes, this makes the roaming Pokémon Lost Forever. The trick is to use the trapping move, pray that they don't use Roar on their first move, and then put them to sleep so you can do some damage to them. Luckily they don't heal from damage between encounters, letting you whittle them down over time. Encountering them at all isn't easy, either - you have to randomly encounter them once before you can track them with the Pokedex, and even then, they change locations every time you change from one route to the next, or enter or exit a city. If you're actively hunting one, you have to jump back and forth over the border between two routes and check the 'dex each time until they're on the same route as you. This was seen as a good idea by the developers.
By the way, in Pokémon Platinum, they have FIVE roaming legendaries. FIVE. Unless you have five Master Balls, a Pokémon that knows a trapping move that works on the legendary AND is faster than it, or a Wobbuffet, you'll probably spend days hunting them down and capturing them all.
Downplayed in Pokémon X and Y with the legendary birds in the post-game. As soon as you encounter one of them (by the way, the one you encounter is determined by which starter Pokémon you chose at the beginning), it will run away without giving you the chance to do anything to it. You will need to encounter the birds ten times before they settle in an isolated cave. They won't run away after this.
Naoya from Devil Survivor, as the first boss fight in Atsuro's Route on Day 7. To win, you have to defeat him. Simple? Heck no, there's a bunch of COMPs lying around, spawning demons unless you picked them up. Then, Naoya can simply throws more COMPs if you take too long. Oh and guess what? Naoya got Devil Speed (7 Movement) and Phantasma (teleportation) to move around.
Vagrant Story: The final boss. SO MUCH. Your only chance of hitting him is when he stops to attack you.
Tela Vasir from Mass Effect 2's "Lair of the Shadow Broker" DLC. She's the only enemy in the game who can use the Vanguard's Biotic Charge ability, which essentially allows her to teleport to the opposite side of the battle area (or to you) every few seconds.
Riviera: The Promised Land has a variation in the Wyrm-type enemies/minibosses, wherein if you let them use a MAX attack, they'll blow YOU away from the battle and several map areas away. As an upside, however, this is the only way to acces certain areas.
The World Ends with You has the raven boss Cornix Canor near the end of week 1. Its battle plan is to stay off of the screen for roughly 90% of the battle, only occasionally flying by while you try to hit it in the second or two that it is actually on the screen. It's actually quite fitting, because in-story, the Reaper who sets Cornix Canor on you is explicitly trying to waste your all-too-precious time.
Dark Sun Gwyndolin in Dark Souls is this. He fights you within an endless hallway, shooting at you from a distance with his magic and arrows. When you get close to him he teleports further down the hallway.
The hallway does actually have an end, which will let you pummel Gwyndolin into oblivion should you reach it, but because of the damage going out from both ends, you'll never be able to do this by accident.
Gin Ichimaru in the Bleach Tower in Bleach The Third Phantom. He gathers his spiritual pressure to increase his strength, and then runs away. He has Five mobility,which for most characters,is the highest they can go. Not to mention his absurdly high evasion,and the fact he has a Menos Grande guarding the entrance. That,and it takes two turns to even get to where he is with characters that have the same mobility. To add further insult, you have to beat him in five turns. Have fun.
Shoot Em Up
Subterranean Animism tosses out the whole "defeatable" part and has Yuugi Hoshiguma sit there onscreen, still throwing bullets at you. While the stage still progresses. Meaning you not only have to dodge the boss onscreen, but the Mooks from the sides as well.
There's also stage 4 of Imperishable Night, which has either Reimu or Marisa as the midboss and boss. It is the ONLY stage for which the boss' music plays at the midboss, and it continues playing as you're chasing them down through a horde of mooks (they don't return to the screen until the boss battle, though.)
In the fangame Concealed the Conclusion Stage 4 has Chen and Ran zipping across the screen, taking potshots at you, stopping only once for a midboss spell card, until you reach Yukari (upon which point they return and start the boss battle proper alongside their master).
In Rez, Uranus (the fourth boss) runs away from the player in his humanoid form.
After Burner Climax has the pseudo-Bonus Boss of Stages 6 and 7, the stealth bomber. In Stage 6, you have to chase after it without falling too far behind, and if you keep up, you must finish it off in Stage 7 using your guns. If you fail either, the game keeps going, but you will be locked out of the last 2 stages (as is the case with failing any other Emergency Order).
Metal Gear Solid 3 gives us The End, a well-camouflaged sniper who only attacks from elevated locations and only moves around if he knows he's been spotted or to keep you from figuring out his location.
In Metal Gear 2: Solid Snake, "Running Man" just runs away as fast as he can and waits for the nerve gas to kill you for him. Chasing him isn't particularly productive, but you can lay mines in his path when he laps you...
Omen Deng from Alpha Protocol. You chase him around some scaffolds while defeating some mooks. Then, when you get to the end of this portion (by using the scaffolds to get around the giant statue in the middle of the room), you get to the actual boss fight with him in a more standard boss arena.
It is possible (though just barely, and only with the right skills) to kill him before reaching the boss arena, which immediately triggers his "run away" script, and makes the subsequent battle child's play (just break through his armor and hit him once for victory). But given how hard it is to do this, even the blatantly unfair boss battle is the better way to go.
In Assassin's Creed: Brotherhood some of the Borgia Captains will make a break for it as soon as they know Ezio is in the area. Some other assassination targets in the series also do this.
The Ceramic Smile in Killer7 screams and runs if you face it. Its weak spot, its heart, is only vulnerable from the front. You have to keep your back turned, then flip around at the last second and shoot it in the heart.
Alternatively, you just turn around and quietly wait there with your gun out. The battle arena is circular, and the Ceramic Smile is an idiot, so it'll just come back 'round to you and skid to a halt while you shoot its weak point.
The vampire that Edward fights in Eternal Darkness books it every time you encounter him until you go after his healing source.
There are two notable bosses in Vampire: The Masquerade - Bloodlines. The first, Bishop Vick, has Celerity and will zip around the room at ridiculous speed while blasting you with a shotgun. The second, Andrei the Tzimisce, teleports away every time you land a solid strike. In both cases, the environments are closed-off and have sparse cover, so you can theoretically stay in one place, follow them around the room, and attack at range... but both have minions (zombies in the first case, fleshcrafted monsters in the second) that will latch onto you, dealing damage and screwing your aim unless you keep moving.
Bach, who warps across a relatively large room (considering previous examples) and shoots the player from large distances with a sniper rifle. Considering that he warps after even the slightest damage, it can get annoying indeed.
Though only a non-optional mini-boss fight, Dead Rising has the fight with psychopath Cliff Hudson, who runs and hides in the service tunnels surrounding the store you fight him in, and only pops out to attack you.
Third Person Shooter
In Burning Rangers the third boss is constantly floating away from you in a donut shaped boss arena. You've got to shoot off its armour before being able to attack is weakspot, problem is if you don't keep up a constant barrage of fire it'll pick up the pieces of armour he's dropped the next time he comes around the circuit.
In Star Fox 64, your first boss on Venom (blue path) is a giant robot encased in stone who runs away while throwing obstacles in your way. Slippy even lampshades this with "Why is he running away?" The boss could probably swat you out of the air quite effectively because it's possibly the tallest boss in the game.
There is also the train on Macbeth that you keep following throughout the entire level. Even when it releases the robot that acts as the boss in the strictest sense, the train keeps moving.
This is rather hilariously used to Star Fox's advantage. One of the ways to defeat it is to hit a number of switches that forces the train onto a track leading to one of the weapon depots. The train is destroyed because, as the conductor screams, the brakes apparently won't work and it crashes into the building.
In Red Dead Redemption one can find the War Horse wandering the fields. The game will alert the player, and the player will be forced to chase it down. Since it runs in any direction it wants, the player has to try their best to keep up. The player also has to be on a horse to even stand a chance of catching it. This makes it more complicated as not only do you have to catch it, but you have to also jump off your own horse and grab that horse right away before it turns away. Killing it removes the chance of getting it, so wasting bullets wastes a good horse.
The Punisher has Eddie Gnucci, who's constantly running around a maze as other enemies constantly spawn. If you track him down, you can get a few shots in before he runs off, and it's very easy to lose him.
Turn Based Strategy
Not strictly a boss unit (although they can be), mounted archers in the various Total War games, particularly Rome. They are the fastest-moving units in the game, and the AI is pretty much instructed to run away whenever any opponent comes close. Even if you have light cavalry, it's unlikely that you'll be able to catch them. Fortunately they are large targets and so are somewhat vulnerable to foot archers, but many Mounted Archers can also form a "shooting circle" and minimize their casualties to incoming fire... This would have been a case of Chasing Your Tail, if it weren't for the Mounted Archers' ability to FIRE AT YOU and run away at the same friggin time. The best way to defeat them is to use the arbitrary "map borders" to trap the mounted archers in some corner... provided you've got enough units to actually trap them at all.
Ruger in Fire Emblem Awakening will start to run away when you get close to him, and is the only boss in the game able to walk off the map. It's not necessary to fight him, but you get a nice wad of money for beating him.
Wide Open Sandbox
Lots of missions in the Grand Theft Auto series involve chases in cars/bikes/anything else, with you having to tailgate your target until he's dead.
A notable one in Liberty City Stories had you chasing a Fat Bastard in a meat packing plant. You are restricted to using an ax and the only way to successfully strike him is to sneak up from behind. If you strike him or are seen he runs to a new hiding spot and is way faster than you.
Taggart in Prototype spends half the time running from you in a Thermobaric Tank and the other half ordering weaker tanks to intercept you. Unlike conventional examples of this trope, it is trivially easy to catch Taggart. Alex runs much faster than the tank. You can even ride the tank to its next stop. The problem is the game won't actually let you finish him off until the plot calls for it (you need to jack the tank to eat Taggart, and that function is disabled), at which point a cutscene gives Taggart a decent lead so he has a minor chance of actually escaping, thus failing the mission.
Hilariously, the distance meter which forces you to stay within range of Taggart's tank is fairly lenient. If you've memorized where the tank ambushes are set up, you can bust them up before Taggart actually gets there.
In the sequel, about half of the Evolved you meet will lead you on a merry chase through the city before finally becoming vulnerable to damage, usually taunting you for being slow even if you have fully upgraded your movement and are thus faster than they are.
In Minecraft, the Enderdragon flies away from you as soon as you LOOK at it. Good luck trying to shoot it with arrows...
When you actually get to fighting Shogo in Saints Row 2, he has a bunch of Ronin lieutenants fight you before attempting a getaway on his motorcycle.
Ultimate Spider-Man abuses this trope. Both as Spider-Man and Venom, you will almost always have to chase the enemy across New York City before finally fighting him/her, either because you have to catch up with the boss (Rhino), because the boss is escaping (Venom), or because the boss wants you to chase him (Electro and Beetle).