The "test" boss or Mini-Boss that appears at the end of the first level, the end of Noob Cave, or immediately after the tutorial. Notable because the player has been fighting nothing but helpless weaklings until now, and this is the player's first taste of "real" combat, in the sense that this is an enemy that actually has a fighting chance to kill you (although the fight is invariably quite easy).
Has a high probability of showing up at the end of A Taste of Power. Sometimes used as the first encounter with the Goldfish Poop Gang. The antithesis of the Warmup Boss is the Wake-Up Call Boss. See also Breather Boss, for when a boss of this level of challenge shows up later in the game and is expected to be harder.
The first boss of DonPachi can be beaten just by firing one charged shot and watching the 'multi assault tank' blow up!
Garland, the very first boss you fight in Final Fantasy I, goes down with little effort for a properly prepared party. His One-Winged Angel form, which you fight at the very end of the game, is orders of magnitude tougher.
Final Fantasy IV, V, VI, and VII all have an initial boss that switches between an attack form and a defense form. When in the latter form, the player is instructed to withhold attacks until it changes back — thus, it serves as a tutorial for the Active Time Battle system that Final Fantasy IV introduced.
Final Fantasy VIII has Ifrit, as a very good example of this. He hits harder than previous enemies, and has a special attack, and likely can't be beat in a direct hit race, without the need to heal, thanks to his higher hit points. But the heroes are given his weakness even earlier in the game, and was taught lessons on how to beat him. So much so, he even speaks special dialogue when his weakness s used against him, to let the player know they are doing it right.
In Final Fantasy XIII, the Shiva Sisters serve as this for the Eidolon battles. These are basically Puzzle Bosses with a time limit where you can't win by attacking alone. You need to use Libra to figure out the strategy that 'impresses' them. Shiva is both easy to impress and almost impossible to die to; one of the Sisters will continuously heal you if your health drops.
The Parasite Queen in the Space Pirate ship in the beginning of Metroid Prime.
Compared to the rest of ''Echoes'', the Alpha Splinter is pretty easy. His first form is actually harder than his second, mostly because its first form's charge attacks are considerably harder to dodge than the second form's, plus the second form gains an attack that is pretty easy to dodge.
In Super Metroid, the first enemy you face is an old enemy, Ridley. Unfortunately, you're at minimum strength and he's a giant flying Space Pirate dragon. Thankfully, he flies off after either of you takes too much damage (more likely you). When you fight him again later, it's a much different story.
Kraid in his Super Metroid and Metroid Zero Mission appearance. His attacks aren't terribly damaging and he dies pretty quickly to missiles. Don't forget that he is massive, and this is compared to Ridley, who is much smaller than Kraid, yet possess more strength and health than him. Doesn't really apply to Zero Mission's hard mode, though.
The Masked Man in Jade Cocoon (Who totally ISN'T Lui, by the way). He's the only boss in the entire game who doesn't summon minions to fight for him, and even facing him just as Levant with the starting equipment he is fairly easy to defeat. Fighting him is optional, but you score a very handy water-elemental spear for the trouble (or lack thereof).
The Darkside Heartless at the end of Sora's dream in Kingdom Hearts I and its Nobody counterpart Twilight Thorn in Roxas' dream in the sequel.
The former also appears at the end of Destiny Islands and as the second phase to the final boss.
At the end of the Birth by Sleep tutorial level in which you play as Ventus, you'll have the option of fighting either Terra or Aqua in a practice duel.
The fight against the Vault bandits in Golden Sun. Despite being potentially able to defeat the party, they're just glorified mooks, and the fight against them was the first plot-relevant fight that was actually winnable.
MOTHER 3 has something like this in the form of the Mole Cricket, the only enemy you fight in the Prologue. Nicely subverted/lampshaded in that he comes back toward the end of the game, demanding a rematch, but his stats are exactly the same.
Aside from having maxed-out speed, probably so that you can see just how weak he still is before your first attack annihilates him.
The Reconstructed Caribou you fight early in the first chapter is also a fairly easy boss, unless you've been avoiding fights.
EarthBound had the fight with Starman Jr., where Buzz Buzz basically did all the work for you.
Every House of the Dead game has a boss with an easily visible weakpoint and revealing movements, so that the player can get the weakpoint system of the bosses down.
This also goes for its spiritual clone/parody Carn Evil. Although sometimes you had to listen to their quotes.
Most of the Spyro games inculde one of these. Toasty in Spyro 1 - you literally have to work hard to get hit in this "battle." Then Crush in Spyro 2 - his fireball attack can be a bit difficult at times, especially when he does three in a row, but one can still easily beat him in less than three minutes. And then we have Buzz in Spyro 3 - for the first half of the battle, he can't even really hurt you unless he knocks you into the lava with his harmless ramming attack, or you mysteriously forget how to run when he rolls after you. You'll be done with him in two minutes - he's so weak that he even comes back later as a mook used by another boss.
In The Legend of Spyro: The Eternal Night, the first boss, the Assassin, is an easy battle that you'll be done with in three minutes tops, and eventually battles with ordinary Dreadwing enemies will emulate this boss.
Sonic the Hedgehog CD's boss in Palmtree Panic is a contender for easiest boss ever that requires some challenge. As in, you can't just attack it whenever. You have to wait for Robotnik to lower his bumper guards. And then after you hit him, you can hit him again pretty much any time. Did I mention he only takes 3 hits to beat?
Also if you run to the right hand side of the screen, you'll be behind him when he lands and can avoid the bumper guards altogether. He can then be defeated in about two seconds.
No More Heroes starts off with the first ranked match against Death Metal. In terms of characterization and hostility he's up there with the rest of the game's bosses, but quite simple to defeat. Lampshaded in that Travis seems aware that he's just a warmup boss and spends half of the battle giving an internal monologue that suggests he's not paying any attention to the fight.
The first encounter with flying mutant toaster Balrog in Cave Story, and if that wasn't easy enough, he even asks you if you want to fight him (and if you say no, he'll run off). You'll meet him again, three or four, in more difficult forms, but compared to the other bosses in the game (e.g. Monster X, the Core or the Doctor), all of his forms are enough of a pushover to make him qualify as a permanent Warmup Boss.
The Mega Man X and Zero series love intro bosses that are both huge and pathetically easy. In fact, the bigger the boss, the easier it is (X2 being the ultimate proof).
The "Rogue Jedi" (actually Kento Marek, the father of the main character) serves this kind of role in The Force Unleashed.
Garam from Battle Clash and its sequel, Metal Combat: Falcon's Revenge. Even though Guido's got a souped-up version of his mecha in Metal Combat, a few well-placed Charge Shots (as few as two if you're playing as Carol) can bring him down in less than ten seconds.
The random bandit leader encountered in the first mission of Fire Emblem 7 (the first one released Stateside) definitely qualifies. It's a tutorial chapter, he is at a disadvantage in the weapons triangle, and the RNG is even rigged to let you win and get a specific set of stat gains upon leveling up.
In the first three Mario & Luigi games, Bowser is the very first opponent you fight (in the case of Partners in Time it was Baby Bowser), and you learn the bare basics of battling while fighting solo (Luigi is only at the sidelines at every occasion). In Dream Team, Antasma takes this role. He goes down in two attacks and he can't hurt you. Justified as it was a dream of Luigi's.
Junior Shrooboid was the Warm-Up Boss for Adult Mario in Partners in Time.
The Dobkeratops at the end of the first stage of R-Type is a pushover. Despite its size and frightening appearance, its attacks are easy to dodge, and it can be taken down in mere seconds.
Devil May Cry 4 uses the series' main character, Dante, as a warmup boss for the new playable character, Nero.
Subverted in the 2nd fight with him, where he becomes harder to beat, and even more in the harder difficulties, where he has full access to his arsenal of weapons and will gladly use them in BOTH battles, even when he doesn't acquire the majority of those weapons until later in the game, thus turning him into a That One Boss. He also becomes a Wake-Up Call Boss to the players that come from Devil Hunter Mode into Son of Sparda Mode thinking that he'll be as easy as ever in the first mission, until he starts using weapons like Pandora and Gilgamesh to attack you.
Beyond Good & Evil throws you into a boss fight against a DomZ monster, after fighting off several of its flunkies. That teaches you about on foot combat. Not half an hour later, it throws you into a boss fight against a flying serpent, which teaches you about hovercraft combat.
Bartholomew Kuma, the first boss of One Piece: Unlimited Cruise is kind of the first boss version of the Anticlimax Boss. Anyone who has read or watched One Piece knows how much of a scary Badass this guy is, and so when the fight actually starts, it's a surprise that the big guy turns out to be a complete pushover. Once you unlock him in multiplayer though, he regains his Badass status.
Smoker plays this role in Unlimited Adventure, and is similarly toned down from his fairly powerful canon incarnation.
Regardless of which route you take through Corneria in Star Fox 64, the level will end with a really easy boss. The one at the end of the Mission Accomplished Route is the same one as the first boss of the original game.
Rollanratl in Wario Land: Shake It! is a Warmup Boss for anyone playing the game, with relatively easy to dodge attacks, and his main attack doing no actual damage. As is Spoiled Rotten in Wario Land 4, except when you have to kill it in 10 seconds on Super Hard mode.
When it comes to minibosses, the Kirby series has Mr. Frosty. His battle tactic consists entirely of screaming and running forwards. He always trips after running for a while. After getting up, he fires an incredibly-slow block of ice. Then, he restarts the pattern. For real bosses, there's Whispy Woods, who is defeated by his own apples.
Whispy is the first boss in almost every Kirby game.
For added fun, to make this fight absolutely impossible to lose, obtain and hang onto the flame sword. Draw the sword and turn around so that the blade is touching the N-Z. Death in about five seconds.
Kisuke's first boss in Muramasa: The Demon Blade seems like this at first, until you deplete his first life bar... And realize that he has about four or five more.
Momohime's first boss more or less plays this straight, though.
Battle Garegga's first boss, Nose Lavaggin, is a bomber that can be easily defeated by continuously shooting at its center, especially with a ship that uses piercing shots. However, doing this instead of destroying the boss's many components means you miss out on hundreds of thousands of points that could contribute to an early 1-Up.
The Ogre performs this duty again in Dragon Age II, but those who played the first game would remember how to fight them, so it's potentially not as bad.
Similarly, Mass Effect 2 throws a YMIR mech at you in Freedom's Progress; it later goes on to turn up as a particularly tough enemy.
Art of Fighting started you off against Ryuhaku Todoh, an aikido practitioner with average strength and speed and exactly one special attack.
The first gym leader in every Pokémon game could be considered one of these. Since the first trainers you encounter beforehand tend to have extremely low-level bug Pokémon that can be beaten without much effort, the first gym leader is typically the first battle in which you're required to utilize strategy and type match-ups. However, if you chose the wrong starter Pokemon or rush in without having leveled up much, it could easily be a Wake-Up Call Boss.
The Magikoopa fight of Paper Mario qualifies, as it comes immediately after you get the Action Command function.
Even earlier than that — even before you get Action Commands — you fight Jr. Troopa, though it basically just boils down to "trade jumps back and forth".
Blooper, the second boss of the same game, also applies to this trope. It is also the first boss to have a chance to beat the player.
Final Fantasy III's Land Turtle can be beaten fairly easily by equipping the gear you find in the cave and using Arctic Winds on it. He's slightly trickier in the DS remake, when Luneth fights alone, but is fairly simple; the Antarctic Wind will still chop off most of his health.
Petey Piranha in Super Smash Bros. Brawl: The Subspace Emissary. His attacks are slow, easily dodged, obviously telegraphed and he has three gigantic hitboxes. You also fight him as Kirby, which means that you can basically just get up in his face and deliver a continuous stream of pummelings during the fight. Once you know his painfully predictable pattern, it's very easy to not take a single percentage point of damage during the curbstomping you give him.
The Wood Man Guard and Shadow Fang are your introductory bosses for Vindictus, if you didn't fight the Giant Spider at the end of the tutorial stage.
The Fists of Grudge in Catherine, the first boss is relatively easy, not very fast, easy blocks to move, and the only special move she has is the ability to change blocks into very heavy blocks you can still move, just very slowly. Which can be undone with a powerup in the stage, no less. After this though, this being an Atlus Game, the bosses get much harder.
In Baten Kaitos: Eternal Wings and the Lost Ocean, the first boss fight is the laughable Saber Dragon in Moonguile Forest. Even if you haven't gotten the hang of the card-based battle system, the dragon doesn't hit hard at all and has a crippling weakness to light and water. Granted, almost immediately afterwards it's followed up by the Lord of the Spring...
The Mad Bomber from Popful Mail. It has an easily predictable pattern, and takes only 10 hits to defeat with Mail's basic longsword.
Every Stage 1 boss in the Touhou series. They're the first spellcard users you encounter in the games, but their patterns are (usually) trivial to dodge. This sometimes extends to the Stage 2 boss, at least when they're not the Wake-Up Call Boss.
In World of Warcraft, the bosses in Ragefire Chasm, the first dungeon, have no noteworthy special abilities, and are only distinct from the trash by having more health and doing more damage. It allows people to ease into their roles in the dungeon.
In the Brawler's Guild, Bruce, the first boss, is a crocolisk with relatively little HP and only one deadly attack, which can be avoided simply by running away from him (people who die to this attack are often considered Too Dumb to Live). There are simpler fights (such as King Kulaka and Unguloxx), but they mainly test the player's ability to put out damage while staying alive.
Wyzen in Asura's Wrath has the easiest patterns in the game to look out for, along with being the first of Asura's former allies to die. Even when he's the size of the planet, Asura still destroys him handily.
Nine-Toes in Borderlands. Even first-time players tend to finish him off pretty quickly, though his pet skags can be troublesome.
Skeleton T in Puyo Puyo is typically this. He's slow and doesn't even rotate his Puyos.
Terranigma has the several early battles against Huballs or Cadets, which barely even count as bosses.
At the end of the first section of the game (in the last of the 5 towers), Ark faces off against the first real boss, Shadow Keeper (a giant demonic scorpion...thing). Depending on how much the player has levelled up at this point, Shadow Keeper may be anything from a Wake-Up Call Boss to a Warmup Boss. If your player is say, level 5 or below, he's a Wake-Up Call Boss (and will most likely kill you slowly). If you're level 6-8, he'll be regular boss. If you're level 9 or more, he's a Warmup Boss who you'll easily trounce.
This extends through the whole game, in fact. As long as you keep Ark sufficiently levelled up, it's possible to make sure that each and every boss in the game is just a Warmup Boss. But don't do enough Level Grinding and they'll each be a Wake-Up Call Boss instead...
The first Homeworld has a variant: where the first level is basically a tutorial and the second level is embarrassingly easy, the third features four assault frigate that you have to defeat while capturing at least one before they destroy all the cryo trays. While destroying them all is rather easy, they still have a chance to destroy all the cryo trays, and if all cryo trays are destroyed or you don't capture at least one frigate you lose.
In Darksiders II, the first real boss, the Crowfather shapeshifted into War, uses easy to avoid though somewhat damaging attacks. It's a good thing they are easy to dodge too, since the fight also demonstrates to those still used to playing as War in the previous game that Death is faster and more fragile than his brother.
In The Addams Family, the earliest boss you can fight is a giant bird on top of a tree, and it's the easiest boss. You just jump on it when it swoops down and avoid the eggs it drops. Like every boss (except the final boss) defeating it earns you a Heart Container.
Diablo III has two of them, the first being the necromancer Jondar, who you fight with the help of your first companion Kormac; and the Skeleton King, the first proper boss of the game.
Shadow Yosuke from Persona 4 is your introduction to general bossfight 101, involving using attacks and guards in response to the boss's behavior to bring him down.
Heavy Weapon has the Twinblade, a Future Copter with two homing missile launchers as its only weaponry... both of which can be destroyed, along with the missiles they launch. In the rematch against it, it gets two plasma cannons that can't be destroyed.