~Dullahan (Golden Sun 2 and 3). The last boss you'll face in Golden Sun, he's designed as basically the ultimate challenge, putting everything you know to the test. He is not easy. I don't think I've run into a boss, in my years of playing games, that is legitimately harder than he is. Star Magician is somewhat similar, though on a lower tier. He's slightly easier in GS3, but not by much.
~Saturos (Golden Sun 1). Not only does this guy have one of my favorite musics ever, he's my favorite by virtue that he is a difficult boss you face in the early game, plus he's an important plot character. He hits hard and by the time you fight him (your party's at level 11 or so) you don't really have access to a lot of healing spells - plus a few of his moves are multi-target and your healing will only be single-target at this point, so it does take a bit of luck to actually take him down. (The Stealthy Scouts in GS3 are kinda like this too, except they lead with Psy Grenades to drop all your party's PP to 0 and then spam status ailments - their HP is kinda low, but they make it so it's very difficult to counterattack.) His battle theme got reused in The Lost Age as the theme for Star Magician, and in Dark Dawn as the theme for
every superboss. He is one memorable son-of-a-gun.
~Lundgren (Fire Emblem 7). Lundgren is the first time you hear the Rise to the Challenge boss theme, and it really gets your blood pumping. And for good reason - he takes some tricks to kill, given how he can OHKO (or come close to) any of your characters not named Sain, Kent, or Wallace. They really drive home the point that you will need
all your party members to team up to take him down. Unless you want to be cheap and use Nils + Lucius over and over, but then you miss out on the dialogue.
~The Arena (Fire Emblem series). Your way of level-grinding. Your character dies and you lose him forever, thus forcing you to restart the chapter and redo all your hard work (except in emulators lol). I've noticed that, if anything bad can happen to you here, it will.
~Giant Moldorm (LoZ A Link to the Past). Seriously, I hate this guy. Have fun getting repeatedly knocked off the ledge while trying to hit him 6 times. I think he was the only legitimately hard point of this entire game.
~Bass Omega (MM Battle Network 3). I swear, BN3 had a reputation for having some of the hardest bosses - and cheapest strategies - in BN's history. Star Force was BN's spiritual successor, and although it's had its fair share of hard bosses (like Apollo Flame, and then later giving everyone DX versions in SF2 and Omega versions in SF3), very few really come close to the sheer insanity that Bass brings. 3000 HP, the most for any enemy in the game, but that is not the worst part. The worst part is he has a 200 Aura (can only be destroyed if hit with an attack doing 200 damage or higher in a single hit, upon which it dissipates) that regenerates after
every attack, unless he is interrupted in the middle of the attack. This means you are going to have to burn up a powerful attack (or use Northwind, which comes in weird codes) to get rid of his shield, and then hit him with something while he's attacking to prevent it from going back up. It is not easy. Or, you could take the
easy way out like I did. ...I'm so lame.
~Duo (MM BN4) and Omega Zero (MM Zero 3). I don't want to say much about these guys, but suffice it to say they are very strong and tricky bosses that came from games I hate. They are final bosses that are legitimately difficult and much more so than the bosses that came before them, something that is really uncommon, so I give them that.
~Whitney (Pokemon GSC/HGSS). I shouldn't have to explain this to you guys. Other notable examples of hellish Gym Leaders are Brawly (if you don't use Sableye), Norman, Sabrina, Roark (have you seen that Cranidos of his?), Lenora, and (debatable) Clay.
~Death Egg Robot (Sonic 2). Man, that thing. That thing. You have to hit it 12 times without getting hit yourself once, and its hitbox is so very small and well-protected that try not to mess up. Heck, I had to look up a guide on where exactly to hit it, and it still took me 20+ tries. The Labyrinth Zone boss in Sonic 1 was annoying for me as well, for a somewhat similar reason.
~Marik (YGO The Sacred Cards). I don't know if you guys have ever played this game or not, but I'll be quick: the deck-building procedure is a lot different, relying on points and card capacity (you can't go over a certain number of 'points' when building your deck), so for the greater part of the game most of your monsters will be relatively weak - heck, when you win Revival Jam off of Strings, it is a godsend. The game itself is also somewhat accommodating of this; the average monster strength of your opponents will remain about that of yours, maybe a little higher. The difficulty does jump a lot when you get to the Battle City finals, though by then you will have a lot more cards to choose from. Marik, however, as the final boss, has the difficulty jump even higher still - all his LV4 monsters have 1800 ATK AND they benefit from the Field Spell that's in effect from the start of the game. At this point you'll be lucky if you can even fit 1 monster with 1800 ATK into your deck unless you've been grinding a lot (and in that game, grinding for deck capacity is slooooooooooooow). And if he ever gets out The Winged Dragon of Ra, he automatically wins if his LP are higher than yours at that point. Either start grinding or start lucksacking if you have this game. If you don't have it, don't get it. You're missing nothing. (I hear its sequel Reshef of Destruction is even more insanely difficult, with bosses that start with 10000 LP or more.) /rant
~Team Meanies (PMD1 Red/Blue Rescue Team). This is kinda analogous to the Saturos fight. It's a difficult boss you fight in the early game and thus you don't have a lot of resources or tricks you can use to really stop them. Worse yet, at this point in the game if you get hit with their poison or paralysis it can be a death sentence. Ekans just smacks you with status, Medicham hits pretty hard, and Gengar does both. It is one insane fight.
~Dialga, or Palkia (PMD2 Explorers of Time/Darkness/Sky). These guys are nasty not only because of their high HP and tricky types (Dialga is resistant to pretty much everything you have, while Palkia is weak only to Dragon), but also their signature moves - Roar of Time and Spatial Rend both attack every opponent, and will do a lot of damage in the process. Dialga's is somewhat stronger than Palkia's, but with the trade-off that it can't attack the next turn - although it will either KO you or your partner, or come close to doing so, necessitating a bunch of Reviver Seeds and/or Oran Berries anyway. Plus, both of them have Intimidator, which might cause your attacks against it to fail if you stand adjacent, making you lose a turn - giving them an opportunity to RoT/SR you again, which they'll be using every other turn anyway. In fact, they basically require you to bring along 6 Reviver Seeds minimum each time you go to fight them just because of that insanity.
That's all I remember/feel like talking about for now. There are some others I've fought (like Shinryu and Omega in Final Fantasy 1, Neo Shinryu and Omega MK.II in Final Fantasy 5 Advance, and even Ancient Devil in Golden Sun 3), but a lot of them fall under a category of one of the other bosses I've mentioned already. (If you're curious, the first two involve a long protracted cycle of attack/heal/rinse/repeat, the next two involve using spam tactics to win, and the last one steals your party members and turns them on you, requiring you to be as cautious when fighting him as you do Dullahan.) Maybe I'll share more, but I think those are the big ones I remember.