Nearly every boss in Devil May Cry does this, especially Vergil ("Don't get cocky", "You are not worthy as my opponent"). When being beaten, he stops this, instead switching to "Now I'm motivated!" and "You are going down!"
Various bosses in Iji will either laugh or make comments. In one case, Iosa banters towards a character assisting you.
Prince of Persia: The Sands of Time features this with the Final Boss. During the fight, both the prince and the Vizier talk to each other. The fight ends once the full run of the speech has been said. However, if you stop attacking, the speech stops, so you can't just casually dodge until it's over.
Quite a few bosses in the Assassin's Creed series. In the first game, Al Mualim taunts you and goes on a Motive Rant in between phases of the fight. In the second game, Rodrigo Borgia taunts Ezio during both fights with him (in Venice and in the Vatican vault). And in the third game, Haytham Kenway goes on a lengthy rant during his fight with Connor about why the Templars will never die.
The Final Boss of Transistor will pause the game just to make sure you're paying attention to what he's saying. In this case it's actually beneficial to not defeat him quickly, as many of his lines reveal quite a lot about his motivation and personality.
The Silent from Lux-Pain. Something is on the top screen in the beginning, near the middle, and near the end for each Silent (though each one says different things). The first and second thing are similar and are related to evilness, while the last one is about it dying. In addition, the Original Silents say a bunch of stuff after you win.
Beat Em Ups
Every single boss, as well as every gang member, in River City Ransom has something snide to say to you as you pummel each other into currency.
The X-Men arcade game has some of the hammiest one-liners delivered by Magneto. "Kill you!" "X-chicken!" "You are DEAD!"
Street Fighter IV has "Rival Battles", where you fight with someone who is somehow tied into the story of your currently selected character. As the fight goes on, voice clips of the two taunting each other will play, sometimes with new clips replacing their usual Super or Ultra startup/finisher soundbites.
Similarly in BlazBlue, characters have altered voice clips when they fight characters deeply involved in their story. These clips vary greatly in nature from each matchup, from the disturbing exchanges between Jin and Ragna, to the hilarious interactions between the ambiguous Taokaka and Boobie Lady, as well as the almost heartbreakingly tragic exchanges when Litchi fights her fellow ex-scientist and former colleague, Arakune. Nu's voice clips change A LOT when she fights Ragna, and in Continuum Shift so do Hazama's.
Shao Kahn from Mortal Kombat will refrain from the pummeling just to inform the player that "You will never win".
Every character does this in Punch Out Wii, corresponding to their moves.
In Jedi Academy Trilogy, the enemies have combat taunts. The dark Force-users are more common to spam their comments, as the lesser mooks tend to die much more quickly.
SHODAN from System Shock talks to you constantly. Constantly. For a self-proclaimed cybernetic goddess, she sure spends a lot of time repeatedly asserting how superior to you she is...
Heinrich I from Return to Castle Wolfenstein constantly brags about how he'll crush you like an insect, wrap you in your own guts, and establish the reign of terror over the world. Earlier in the game, Deathshead provides a sort of this — he's gloating in the control room while you're fighting the Ubersoldat below.
"Fools! Killthe onein the dress!". Especially the dear, old Kael'thas Sunstrider in Tempest Keep. He's like he can't do nothing but talk, talk and talk.
In the Wrath of the Lich King dungeons, it's sometimes hard to find a boss that will shut up for more than 10 seconds. However, this crosses often into Calling Your Attacks, which can be useful, or Bond One-Liner, which can be pretty awesome.
RuneScape: While 99% of the game's dialogue is read-only, several of the bosses of daemonheim have voiced banter. Laughing Mad Blink, in particular, who says the most ridiculous things when you fight him; 'M-More tea, Alice?'. Aptly-named Yk'Lagor the Thunderous was the first boss with voiced dialogue, and ALL of his banter is in the form of Thundering One-liners: "This is... TRUE POWER!""Another Kill For The Thunderous!!"
Foxbat in Champions Online is rather notorious for this, including lines like "Admit it, you're just here for my autograph!" and "Would you classify yourself as a creepy stalker, or just an obsessed fan?"
City of Heroes not only features this frequently, but also allows it as part of the player-created content Architect System. Bosses can react as their health drops during the battle. Depending on the boss, this can be epic, hilarious, or annoying.
Even though the exact line in question wasn't delivered by the boss, Shadow the Hedgehog features a rare example where the in-battle banter is highly important to the plot: The entire game's branching storyline is about determining what really happened to Shadow during his memory gap and whether he is still the original Shadow or a duplicate, and the various alternate endings provide contradictory revelations. During the true final boss battle, the supporting characters periodically give banter that doesn't repeat, usually updates on their current situation, but waiting long enough (longer than it usually takes to win the battle) results in Eggman revealing the truth: one of his robots saved Shadow after the events of Sonic Adventure 2, and though he suffered from amnesia, Shadow survived. This is the only place where the truth is ever revealed.
Gruntilda in Banjo-Kazooie and Tooie. Particularly in Tooie, where she asks you quiz questions during the final battle (and goes easier on you if you get them right). Most bosses in Tooie speak during the battles as well.
Chorus: Who are you, Mastermind? Mastermind: Who am I? Come get me and find out!
In the 2009 version of Bionic Commando, the creators play around with this a bit. If you fight one boss enough times by losing, the pre-battle banter lampshades a Mondegreen. Then there's the penultimate boss, who rants to you in between attacks for awhile. Lose to him enough times, and you get a "skip dialogue" button which manifests as Spencer telling the boss to shut the fuck up and fight.
GLaDOS in Portal spends the entire final confrontation telling that you're a bad person, that things will be much worse if you destroy her, making petty insults, and being bitter and resentful. For that matter, the whole game consists mostly of Boss Banter, with GLaDOS slowly dropping the facade of being a helpful AI.
Some demons in Nethack. Examples: "I first mistook thee for a statue, when I regarded thy head of stone." "Go play leapfrog with a herd of unicorns!"
Real Time Strategy
Command & Conquer Generals: Zero Hour has a "challenge campaign" that puts the player up against the various subfactions in the game. The leaders of these subfactions taunt the players and react to situations on the battlefield. "Oh, look! I have a nuke all ready for you!" Some of these taunts seem to play randomly, though. For example, a futile attack of around three helicopters against a heavy anti-air defenses is accompanied by the enemy General noting that "Fools rush in, General."
All the stronghold missions in Dark Crusade have the leader characters of the factions perform Volleying Insults with each other as the mission proceeds (except for the Necrons, who are The Voiceless and so have a human recently converted into a Necron do the bantering for them). Most of the them are pretty generic "Prepare to Die!" "No, You!" type of things, but some of the insults get incredibly personal; Taldeer and Eliphas for example pull off some epic burns against Captain Thule, while Space Marines vs. Imperial Guard is a rather tragic (for the setting) example of Just Following Orders.
All bosses in the Dawn of War II campaigns that aren't Tyranids have a few things to say to the player, and as usual the Orks have the best lines ("Me only regret... is explodin"). In the case of Kyras in Retribution he customises some of his responses based on which faction the player is using.
Richter: What do you know?! My methods are my business! I don't need you telling me what to do!
Tales of the Abyss ramps this up by giving nearly every human boss a conversation sequence at the start of the fight... which changes slightly depending on which party members you're using. The Final Boss even changes more depending on who's leading the party.
Tales of Xillia has this, though with the addition of stylish animated character portraits that appear as each character speaks their lines.
In Paper Mario most of the Boss Banter follows a progression, and they tend to only say something when entering a new phase and when you discover a weakness. Although the Koopa Bros. alternate between two units of chatter while they're hidden in a lame Bowser disguise, and later, when they're out of it, say "Woah! We're going over!" when Mario throws them off-balance.
The Final Bosses of the series are the real banterers. Especially the Shadow Queen, who, when she becomes invincible, will taunt you until you attack her three parts before killing everybody in your audience to replenish her HP.
The most prominent of these is probably Rawk Hawk — a grandstanding Professional Battler who is in it for the show.
In Mario & Luigi: Dream Team, a few bosses do this. Popple calls you an idiot and proceeds to annoy the boss he's fought alongside in the process, Earthwake has a seeming meltdown at about one third of its health and talks about how badly its been damaged, and Pi'illodium mocks you every few turns or so in Robo Speak. Amusingly, the latter actually changes its quotes when its losing, going from bragging about how your end is nigh to worrying about its own gradually worsening condition (and then going right back to bragging when it heals). The other giant bosses do this a bit too, with the Zeekeeper commenting on the battle as it fights you and Bowser talking about finishing you soon.
The end bosses in Baldur's Gate tend to do this. Not as much as some, and it mostly involves shouting battle-cries, but still...
Common in Persona 3 and Persona 4, though in the former it only appears when fighting Strega, the last boss, Elizabeth, and your teammates in The Answer; the other bosses, being Shadows, don't speak human languages. In the latter, every plotline boss does it.
Gilgamesh from Final Fantasy V has a tendency to talk more than what he actually fights... frequently giving excuses for him running away and giving advice to each of your party members before his Heroic Sacrifice.
Final Fantasy X is a rare example where the player can respond to the taunts. During certain fights, a "Talk" option appears on some characters' combat menus. Using it wastes a turn but grants the character a rather large stat boost or, in the final boss fight, calms the boss's anger and thus his Overdrive meter.
Final Fantasy XIII: FULFILL YOUR FOCUS AND GAIN ETERNAL LIFE. DESTROY THE WORLD THAT DESPISES YOU!
IGNORANT SLAVES OF PULSE! DOES ETERNITY NOT ENTICE YOU? THEN YOU CHOOSE THE PATH... OF OBLIVION!
Dagoth-Ur in The Elder Scrolls III: Morrowind spends a lot of time during the Final Battle explaining why you cannot kill him: he is a god. It turns out, you really can't kill him until you destroy the source of his divinity.
This is a tradition in the MOTHER trilogy. Each of the game's final bosses interrupt the action every turn to say something, and then attack.
MOTHER 1 has Giegue's Reveal that Ninten's great-grandma Maria raised him, and how George stole vital information from his planet that could be used (but, unknown to Giegue, never was) to betray his people. He then pulls the You Fool! card and brags that they cannot hope to stand against him. He then lets Ninten, and only Ninten, know that We Can Rule Together, and when Ninten refuses, he stops bantering. When they sing Maria's song, he screams at them to stop.
MOTHER 3 has two: Porky brags about how you can never defeat him and his immortality (simultaneously implying Who Wants to Live Forever?). The Masked Man doesn't talk himself, but Hinawa interrupts, begging the brothers to stop fighting.
All of the Organization XIII bosses have some pretty cool lines during their battles. And then we have Quinton Flynn's wonderful work with Axel, which, to sum all of his lines up: "BURN, BABY!"
The Final Boss of Kingdom Hearts II has some, uh, memorable banter. First, for those of you who haven't played the games: It is, among other things, the story of two boys who are close friends and, before the game started, rivals, who, after making some horrific mistakes and serious sacrifices, spend three games and the better part of two in-verse years trying to fix what went wrong, find each other, and go home, their rivalry abandoned because dammit, there are more important things than one-upping each other, and through it all, never losing trust in each other. And all that is if you don't believe they're romantically inclined towards each other. The final boss's taunts?
"Sora... are you certain you can trust Riku?"
"Riku... are you sure you're not jealous of Sora?"
Sephiroth has "Descend.... Heartless Angel", "Give in to the dark!", and if you stray from him without fighting for too long... "What's wrong? Are you afraid?"
The final boss of the original Kingdom Hearts. Due in part to both awesome dialogue and an awesome voice actor, some of the game's most memorable lines are spoken by the Big Bad during the battle with him.
Saix says some of the most badass lines while trying to pummel you.
The Witcher is filled with banter between Geralt and named enemies. The most is with Azar and the Professor because of their history of chase and false finishes, with banter before, after, and throughout combats.
Azar Javed: Did anyone warn you about pissing upwind?
Geralt: You talk too much.
Azar Javed: You just pissed in a tornado!
Drakengard 2 does this often. Of particular note is Yaha, whose banter consists partially of him HITTING ON the male characters. He's also male himself, so...
In Destiny Of An Emperor, Sima Shi and Sima Zhao keep taunting the heroes as you beat the stuffing out of them.
Several boss battles in the Knights of the Old Republic games will pause partway through so you can have a brief conversation with the boss. Sometimes, you can convince them to stop fighting you, maybe even join your party, but when you fight Darth Malak, the best possible result is that he admits that he made a mistake going down the dark path before he dies. All the other results just end up with him taunting you, and most of these conversations start out as taunting and Volleying Insults.
Harbinger's battle quotes during Mass Effect 2: “This hurts you”, “You cannot resist”, “This is what you face”, “Face your annihilation”, “You are bacteria”, “You are shortsighted”, “Pitiful", “We will end you”, “We are the beginning, you are the end”, “I sense your weakness”, “Your attacks are primitive”, “You cannot sustain your attack”, “Your attack is an insult”, “You are no longer relevant”, “My attacks will tear you apart.”
Kai Leng in Mass Effect 3 tries this out during the final battle with him. It backfires.
Xenosaga bosses frequently taunt, curse, or threaten you both in person and over the comm. The second game's Final Boss has Albedo spend most of the battle simply taunting Jr. Justified because he wants his brother to finally kill him.
Many Resonance of Fate bosses find deadly combat to be the best time to exposit their political and religious philosophies.
All the human bosses in Eternal Sonata will throw one-liners at you when the battle begins, before calling their attacks, and once their turn is over. Special credit goes for the last two bosses, who spend longer talking that performing their attacks.
Bomberman Quest, a comedic RPG spinoff from the main Bomberman series, had this set-up: before every battle (whether with a boss or a mook), Bomberman and the enemy character would have some humorous back-and-forth banter; even the True Final Boss does this, except his banter is completely serious.
Shin Megami Tensei IV actually allows you to respond to the banter, with your choice affecting the battle - a boss may feel intimidated or disgusted, lowering stats or eating his Press Turns, or you might gain stats as you get into the fight. Miss the correct choice and instead the boss will power up, or you will start losing heart. In the last fights, it's very important you actually think the available options - actually picking the answer that does not help you might well spell doom for you and your team.
Inverted in The Last Story, where it's the main characters who continuously discuss about the possible weaknesses of the bosses they fight. There are a few exceptions, such as human (or humanoid) bosses who can speak.
Every major boss fight in Time and Eternity engages in this and is the source of much of the game's humor. The only tricky part is keeping the fight going long enough to allow everyone to finish speaking.
Shoot Em Ups
And, of course, Ultimate Crab Battle. "You cannot stand between me... and my destiny!" and so forth.
Most bosses in Star Fox 64 talk during the battle. Surprisingly, Andross himself mostly talks in cutscenes and contacts you during the final level to taunt you, but during the battle, he mostly laughs. The Star Wolf team is especially prone to do this, with Pigma having some pretty cruel banter, such as "Too bad Dad's not here to see ya FAIL!" and the surprisingly harsh (for a Nintendo game) "Daddy screamed REAL good before he died!", both referencing how he helped conspire to kill him.
Touhou usually restricts boss dialogue to before and after the battle, but some levels feature an exchange with the boss in the middle of the level. Ten Desires takes it further, with one spellcard of the final boss being her calling for assistance from her two allies, all three of them with dialogue boxes above their sprites, the battle not interrupted whatsoever.
Every Metal Gear game from Metal Gear Solid onwards loves this trope — from Ocelot comparing Snake (favorable or not depending on how long you take) to Big Boss, while The Fury will scream insults at the player when he gets hit, etc.
In the first Hitman game, the drug dealer (read: a living Shout-Out to Scarface (1983)) will interrupt the battle for each and every bullet he gets hit with... even when you're using a machine-gun.
In the final level of Thief II, the Big Bad is aware that Garrett is somewhere in the building, so periodically speaks to him over the loudspeaker, taunting him and trying to get him to give up. Garrett is too smart to allow this to affect him, only muttering "Yeah, yeah, keep talking" under his breath at one point.
Just about every boss in John Woo's Stranglehold will do this during your battle with them.
In Resident Evil 5, every boss-fight with Wesker involves him taunting you and/or explaining his evil plan in great detail. Towards the end, Chris finally had enough...
Wesker: The human race requires judgment.
Chris: And you're going to judge us? Do you get all your ideas from comic book villains?
In Resident Evil 4, the boss battle with Salazar is the ONLY boss battle (besides the final boss) to talk with you. And it can be used very tactically, as he only talks for 3 reasons, besides his introductory "You will suffer like Ashley did". 1, whenever you shoot the Verdugo, he mocks you saying things like "That tickles"; 2, whenever he grabs with you with the tentacle, and the best of all, when he uses his one hit ko move. Which is very useful, since you can tell when to dodge.
The Marker: YOU WILL ALL BE MADE ONE!MAKE! US! WHOLE!
In Gears of War 3, the final fight with the Queen is filled with this.
The bosses in Syphon Filter do this. When they go off at Logan's feelings for Lian, it's pretty effective.
In Kid Icarus: Uprising they don't just talk throughout the boss fights, but the entire level too. In a World of Ham, this can be pretty entertaining — and distracting on higher difficulties. Those that don't speak themselves result in chatter with their rulers/manipulators (Cragalanche, the Underworlder rematches, Phoenix), between Pit and Palutena (Twinbellows, Great Reaper), or even between Pit and Pitnamely, Heart Of Hades.
Quite a few bosses in Fire Emblem will say something along these lines when you first attack them, or when a specific character attacks them.
Super Robot Wars in all of its incarnations does this. Oftentimes, this only happens when the current boss you're up against has some relation to the person that's attacking them, but the final boss of a game almost always has a few lines of conversation with any conceivable member of the group. Banter also accompanies literally every attack delivered, received, dodged, or intercepted. (Warning: Video contains spoilers in the above-mentioned variety of Boss Banter.) And even that is sometimes preceded by unique one-liners when fighting bosses.
Final Fantasy Tactics - Quite a few story battles consist of Ramza arguing with his opponents, trading dialogue (and/or banter) at various (scripted) points of said battle, turning these fights into an exercise in Shut Up, Hannibal!. If you take along the right special characters to the right story battles, they've got some optional dialogue too.
Sasha from inFAMOUS keeps talking about being in love with you and being angry about you loving another woman while you fight her. She at one point mistakes you for Kessler, foreshadowing The Reveal at the end. Kessler too, made funnier or more sad by the fact that, as he is Cole, he knows exactly what to say to make him feel bad.