I'm dissapointed in how no one yet has mentioned a single SMT boss. Closest to that was ILoveNima mentioning Mega Dark Angel Hippie from P4, and he's pathetically easy, just a damage sponge (harder than Cabagge Boy though). How can you talk about hard bosses without mentioning Atlus's One Franchise? I may bring one later but for now, I'll start with a classic:
Digital Devil Saga 1, the Demi-Fiend. You see, this guy, is one of the toughest RPG bosses in history, and it's not an exageration. You must have at least nearly everyone at Lv99, with the best equipment and skills, just to have a small chance to win.
He has physical skills that hit your whole party multiple times, can cause status effects, and hit hard individually even when your defense is buffed. He is inmune to physical and elemental damage, so you can only hurt him with costly Almighty skills, and even then you won't deal much damage. Oh, and he has Pierce, so Phys resistance is useless. And if you got something that makes you inmune to his attacks somehow, then he uses Gaea Rage from the get-go.
He has like 10.000HP, and when he's about to die one of his Demons will use Diarahan to bring him back to full HP (thankfully, only does that once). When the battle begins, he summons 2 Demons, Cu Chulainn and Grimekhala, who each is a royal pain individually: Lancer increases your attack but heavily drops your defense, which means he'll kill you in seconds, and when not doing that Lancer will deal heavy Phys damage to you. Cyclophant meanwhile is constantly spamming Mamudoon, insta-kill party-targeting death spell with high chance of working.
And if you kill one of them he summons another Demon with different, but equally or even more annoying strategy. And every 3rd Demon he summons, he uses Gaea Rage, which insta-kills everyone in your party and it's almost impossible to avoid.
There is a strategy that makes beating him easier and even can avoid GR. But, it involves a specific character (the worst, actually) and a specific skill, and the best part? The strategy itself is heavily reliant on luck to work, and after that you still may lose. Videos about people beating him usually last about an hour. And even after defeat, he still manages to be badass.
The fun part though, is that Demi-Fiend is the protag of SMT3. In his game, you can get all the skills he uses in DDS….which are all mediocre, early/mid-game skills. Even his Demons are mid-game, with 1-2 exceptions. Hell, his boss fight theme in DDS is his normal battle theme in SMT3.