I thought that the Enderdragon from Minecraft was annoying and took ages to kill. You first have to destroy all the towers which takes almost an hour. Then you have to destroy the dragon itself as it's flying. Plus, mobs spawn on the surface so it's even more difficult.

One of the most annoying bosses I have ever met is definitely Berserker from Legend of Legaia, It can inflict Venom (decreases your HP every turn.... -.-) and Rot (immobilizes one of your limbs, very very annoying..) And when it's HP starts to get lower, It starts to use Stone Circle attack repeatedly which can outright kill the entire party if they are not using Spirit (Defensive mode that also raise your AP.)

Most annoying boss that I have done is probably Shao Kahn 2 or 3 hits and your beat and he uses the same attacks over again and to make things that bit more annoying he even taunts you throughout the fight and the amount of retries is limited

iggy koopa in super Mario world I hate him sometimes I miss a jump and get hit by that stupid guy and the platform Mario and iggy are on keeps tilting back and forwards its just really annoying. His pretty hard for the first boss of the game

Crunch from Crash Bandicoot: The Wrath of Cortex was annoying, especially when he had the help of Pyro. Dayum, he was hard to beat as a little kid. He won't stop popping up.

I haven't played the game since I was little, but i'd have to say more than likely it'd be Sephiroth in Kingdom Hearts 2, during that little special encounter that people normally run away from, he's the only boss I never managed to beat in a video game, but I know i've seen people beat him on youtube.

Freaking Xaladin from Kingdom Hearts 2! I hate his freakin' wind lance dragon's wind breathe!!

I am not sure if I have posted here before.
 
 I think that eggmans machines in the first 3 sonic games are annoying because same way to beat it every time. Jump on the glass! Another boss I think is annoying is the egg dealer from shadow the hedgehog. You know what they say the more the merrier!

FiresOfAxel : "The Fury" right? I remember that when i used to play it in the PS2, i'd battle him without any problems. But then, two months ago i bought the "Metal Gear Solid HD Collection" for my Xbox. And i couldn't beat him for whatever reason. I forgot right now, but there are some little tricks against him that i didn't remember at first. But yeah, annoying at first. Even the Boss was a lot easier than him ^^

I can remember some bosses:

The last boss in Sonic Generations: The boss is not annoying. It's the level. Have you tried to get the achievement of winning without getting hit? It was hard as ****. It annoyed me a lot, and at first, it almost made me quit the game. I didn't though.

Mr. X, a.k.a the last boss in Streets Of Rage (No, not the RE 2 guy): Back in the day, game weren't made so you could win. They were made for you to lose. And Mr. X is a prime example of it. If you don't trap him you're going to die, A LOT. And if you do trap him, he's still going to kill you once or twice. He was so annoying when i fought him alone, ugh. Bad memories, bad memories.

I'll edit this if i remember another boss.

The most annoying bosses had to be all the final boss in the Xenosaga series and Smough and Ornstein in Dark Souls.

SuperCrash64 : The boss that was annoying to me was orphan on Final Fantasy 13 because he did an attack that would get all your party members near death situation every time he would use it.

Supergamer : Xigbar was way worse in my opinion.

I managed to defeat Xaldin on the second or third try with Mickey and a bunch of strategies easy but Xigbar took me hours! He keeps teleporting everywhere and shooting things at you. His special attack is practically unavoidable and takes away your whole health bar. It's just ridiculous.

Ripper Roo. That laugh is the worst and his design is nightmare fuel. i used to be scared of him when I was little. The final boss from Sonic Generations (I won't say his name for spoilers sake)it isn't the boss himself it is the fact that the rings are not very visible and the constant "you can do it sonic" gets me annoyed.(reminds me of Sonic Heroes. Great game) The wonderful 101 bosses are annoying in that they take way too long to beat. It isn't that I am bad at it. It is because their health bar is very long and sometimes you will be fighting the boss next you'll be chased by it then you'll be playing in the style of Space Invaders then finally you have beaten the boss. The final boss is the worst. That took at least a hour and a half to beat. There is a lot of text and dialogue. Also he has a lot of forms where one time he copies your abilities,one time he turns into a giant mecha machine. Also there is a chase sequence as well as a button mash at the end. I thought my finger would fall off. I enjoyed the boss battle but felt like they were on longer than necessary to me anyway. The Eggman boss battles in Sonic Unleashed. (wii version. I can't remember if this is also in the dragoon boss as well). Egg-man keeps saying "you little" and I can't get it out of my head. I don't hate all Sonic boss fights but those are the ones that I can think of besides Lost World's final boss which isn't annoying but it is similar to Sonic Colours. That is what i think are the most annoying bosses. I didn't think i would write that much. Whoops.

SilverYoshi : To me all the devil bosses are annoying

Wizpig from Diddy Kong Racing was very annoying to me. The first boss race against him was easy he was really big and ran pretty fast. However, the second boss race against him was a tad difficult. He was using an rocket to race and it made time twice as fast. What made it more annoying to race him was the race track. There is literally shooting projectiles everywhere and if that didn't stop or slow you down the race was also designed with a few walls right in the middle of the road. He though you would use an plane to to race against him this things would sometimes get in the way. You have to be an very skilled racer in the game to beat him.

There is the one demon on the bridge from Dark Souls. It was really easy, but it would also easily knock me off the bridge. It was not too difficult, but frustrating when it was really close to being killed. There is also the Phoenix and Griffin double archfiend level in Soul Sacrifice. That boss took me forever to beat, even with my Blood magic. I got really scared when I destroyed the Phoenix's tail, as my blood magic was unable to hit after that, but I also had some stone throw magic, and it worked out all right in the end. That was still an extremely difficult and annoying fight as the Griffin would constantly blind you, and the Phoenix would burn you. 

Kintaro from Mortal Kombat II is probably the bane of my life in that game. He's like Mike Tyson in Punch Out.

And don't get me started on the final boss in Mario and Luigi: Partners in Time, not counting the bonus boss. The final boss takes FOREVER to kill. It took me, what, one hour or so?

And oh, ho, Tabuu from Super Smash Bros is my LEAST favorite final boss. It took me AGES to defeat him. His attacks are WAY too overpowered. And if you die, you're sent back. And this all happens on the worst level in video games: The Great Maze

I remember as a kid I always had a issue dealing with that one guy from star ocean till the end of time the gun guy you fight when you reach the real world he`s practically a damage sponge and he deals no mercy in trying to kill you after a while like 30 mins and your almost done he cheap shots you and you got to start all over again either that or I was severely under leveled for that guy because I remember there being more content after it.

Anyone played sonic 06? No? Good, don't. 
There are many reasons why not to play this game but silver is one of the big ones.

You have to fight him in the sonic and shadow campaign and he can kill you pretty much easily. Because of the s***ty camera angles you can't see him coming for you all of the time and when he does the chances are he will kill you since you are near a wall. Oh I forgot to mention, he is one of those bosses which can be hit while doing something so you go too close he throws you away and says "It's no use", he is very right about that...