With a rich tapestry of gods and monsters from many different mythologies around the world and throughout history, it's no surprise that the various series under the Shin Megami Tensei umbrella have plenty of awesome boss fights to offer.
We can almost forgive Atlus for making them so hard almost all of them are in the That One Boss page. (And the Nightmare Fuel one, for that matter.)
The main series of games in Shin Megami Tensei is full of these, but none come close to Matador from Nocturne. In a game full of hard bosses, he manages to be the hardest overall and that only serves to make the fight much more awesome.
There is also Lucifer, True Final Boss of Nocturne... He has a 75% resistance to every attack (even the almighty ones). He can be rationally defeated ONLY if you have the right skill (Pierce, which you can only get after beating the bonus level)... SMT has plenty of these bosses.
Babel, in Devil Survivor, summoning EVERY SINGLE MAJOR BOSS YOU'VE FOUGHT, and then having to face it as it opens to reveal the demon world... with awesome music to match.
And at least in one, once you've trashed the Bels, one after another, you command Babel to shunt away all demons in the area. Babel complies... including your own demons, so you have to pound an Eldritch Abomination into dust with what amounts to your fists.
Lucifer also makes a comeback. Calling the fight a feast and calling itself your father, summoning some of the most powerful Demonic Spiders of the game, reflecting any and all physical attacks, and viciously, gleefully abusing the evil that is Megidoladyne. Oh, yes. You're goin' down, pal.
And his speech when you finally destroy him. Lucifer himself trash-talking Belberith is a thing of beauty.
Lucifer: Haha... Belzaboul was doomed. No soul should possess your wisdom, strength and beauty... Go, Abel. Belberith will fall before you. The scarlet snake is at your side...
The duel between Gin and Azuma. Particularly if you have the party dispatch his goons, then just sit back and watch Gin kick Azuma's ass. Yes, it is possible for Azuma to be defeated without you intervening—Gin has a very excellent skill and demon setup that allows him to survive on his own—unlike many other NPCs, friend and foe, who don't bother to revive allies most of the time, he makes the smart choice of using the Revive auto-skill!
Shin Megami Tensei: Strange Journey has great bosses. Alilat has Diarahan, fully healing herself at whim, and being capable of firing off Mind Charged Megidolaons for complete wipeouts, and bounces off physical and Gun attacks. Mother Maya insta-kills anybody who dares use magic against her. Ouroboros uses an attack to randomly slap you with any ailment it wants and heals itself constantly in its first phase, then uses a very strong physical attack on the second. Morax, the very first boss, can be a nightmare if it begins using Bobasma, as can be Asura with Asura Roga. Mother Mem Aleph uses intensely destructive magic of all alignments and random physical attacks that are very capable of killing entire parties. Demiurge loves spamming Big Bang (Almighty-type attack, stronger than Megidolaon) and equally vicious offensive magic. And they're all awesome.
Nocturne also gave us Albion. He has a very unique gimmick; his backup are all based on a certain element, and since there are 4 major elements, it's like you're fighting 5 bosses at once. The Zoas will spam -dyne spells while Albion knocks your party around with physical moves. In addition, if you ignore the Zoas and just focus on Albion, he's instantly revived at full health. The reverse is true as well; Albion can revive all 4 of the Zoas at once. He's also a Large Hamin a game with no voice acting.
Albion: LINE UP THE FLEET! TIME FOR SOME HEAT!
Every Fiend is a major nightmare to battle. None of them, however, come close to Trumpeter. In Nocturne, the brutal combo of Holy Melody/Evil Melody and his overpowered attacks make him already nigh-unbeatable without a well-rehearsed strategy. In Shin Megami Tensei: Strange Journey, however, he manages to top himself - still a major powerhouse, he adds Horn of Fate to his arsenal. This means he can effortlessly kill anyone who is vulnerable to either Light or Dark alignments instantly with virtually no chance of stopping him. And he also has Hostile Terror, meaning he can now slap all enemies with any ailment he wants. And he STILL has Megidolaon.
In Shin Megami Tensei IV, pretty much any boss where you can have dialogue choices during the Boss Banter. Such as Minotaur, which allows you to pull off a couple of Badass Boasts, which throw the Minotaur off guard. This is taken Up to Eleven in the Neutral path, with friggin' Lucifer awestruck and Merkabah acknowledging you as his Worthy Opponent at the correct answer.
David shows up right out of nowhere in a sidequest, but his appearance is very memorable. In addition to providing a really challenging fight, he comes with a remix of the Nocturne Fiend theme while ominously foreshadowing how the other Fiends are in the game too.
The two-part boss quest "Rebirth of the Great Overlord". First, the Red Knight, who you saw attempting to summon Baal in the "Mysterious Story of Tennozu" quest, reveals himself to be your former king, and attempts to make a sacrifice out of you. He uses the same moves he did before, so it's not too terrible of a fight. Then comes the true challenge of the quest: Baal reveals himself to be Beelzebub, who is level 90 and has several attacks true to Beelzebub's association with pestilence such as Pandemic Bomb (Sick status to all enemies) and Death's Door (1 HP of Almighty damage, HP to One if the target is Sick). To drive home the point that you're up against the hardest Challenge Quest boss in the game, the music for this fight is not the quest boss music or even the Domain boss music, but the Main Quest boss theme.
Persona 2: Innocent Sin's Final Boss: Great Father Nyarlathotep. The chaos of Humanity's heart incarnate. Extremely vicious attacks, pummeling both HP and SP. Unforgettable boss.
Nazi warmechs created from rumors. Doppelgangers created by the avatar of madness in your image, wielding your powers. Adolf Hitler. Let Nyarlathotep see what it can come up with; it's going to get junked anyway.
Persona 3. Nyx Avatar. 13 forms, no stopping. That's where it gets epic. Where it becomes a Crowning Moment is the battle after that, wherein the main character unlocks his true Arcana, that of the Universe, representing ascension/enlightenment, stands up to a monster which is single handedly destroying the world, takes said world-destroying blast to the face — repeatedly, ignores it, raises one finger and blasts him out of the sky."It's Useless," indeed!
It's even better when you beat the first 12 forms with (relatively) little difficulty, and then, when you think it's all over... "I sense... Death! Nyx's true Arcana is Death!" This is when the fight actually gets HARD, with the last form having a lot more HP than any of the others, specifically, 4x, and the boss uses its most powerful attacks, such as Night Queen, which can screw you over if you don't know what's coming. Plus, the running dialogue during the fight is awesome, especially if you realise that what Nyx Avatar says before turning into a certain form describes what that Arcana represents in the Tarot.
Any battle against Strega. Fighting Jin and Takaya particularly, though there's something of an emotional level involved too when fighting Chidori. Not only are the battles epic on their own, but they have the added value of being set to "Unavoidable Battle", possibly one of the most hard-rocking and epic boss battle tunes EVER. Plus, they are the only bosses (apart from Nyx Avatar) who have running dialogue as you wear down their HP.
For Arcana bosses, the fight against Strength and Fortune stands out. Not because of its difficulty, but more because of the fact that it's easy to manipulate the Wheel Of Fortune attack, causing damage and bad status effects to the boss while you laugh about how it constantly gets hoisted by its own petard.
Also, Bonus BossElizabeth. The only enemy in the entire game (without multiple forms at least) that can SURVIVE Armageddon, and if you pull it on her, SHE KILLS YOU INSTANTLY. And she can also SWITCH PERSONAS just like you, making it essentially like you're fighting yourself. It adds challenge by basically making it impossible for you to cheap your way out of it, since you are not allowed to use a Persona that nulls/absorbs more than one element, so no Nul-All Abaddon for you. Easily one of the hardest battles in the game, even if you manage to get Orpheus Telos. For an added epic feel, you have to fight her alone, making it essentially a DuelMirror Boss.
Persona 4 has quite a few examples, too. In order of appearance, there's...
Shadow Mitsuo, who starts the battle by summoning a barrier that looks like an 8-bit video game hero.
Magatsu Izanagi, who looks like your own first Persona... if you were not such a nice person. And after him comes...
Adachi. Just like Strega before him, he has a Persona, which is an alternate form of the protagonist's starting Persona. Sure, the battle is easy, but managing to hunt down the killer and fight the guy that's caused so much trouble is just so satisfying.
Ameno-sagiri, which boasts one of the best battle themes in the game.
And last but not least, the secret boss, Margaret, who has an arsenal of Personas with which to take you on. That's a lot!
Digital Devil Saga series
The Demi-Fiend, secret boss of Digital Devil Saga. You MUST be at level 99 with EACH character (and have maxed out EVERY single stat.), just to have a 50% chance of defeating him. And if you take even one skill that can give you any type of advantage (like the Monotheist mantra skill) he just uses an instant death attack that cannot be avoided (Gaea Rage) and it's game over.
And for that matter, the few strategies that can actually beat him are rather complex and still require some luck to work.
Fun fact. The developers of the game even had great difficulty fighting this boss, even in debug mode where you can see and edit the game's script.
Jack Frost. He's not fought often as a boss, but hoo boy is he a damn great boss in Digital Devil Saga 2. Taking a page from the first game's Vasuki, he uses his Signature Move Breath to freeze enemies, then shatters them into oblivion with physical attacks. Say you come protected against Ice attacks. Fine, then. Eat Megidolaon, loser.