On design, fairness, difficulty, fun and flashiness. I'm reminded of many, mostly from bullet hell games but I've found one that is flawless on absolutely all fronts. Sword Saint Isshin Ashina. I wasn't paying attention to how well designed and expertly crafted this boss is back in 2019 cause I was pissed off at dying so much. The fact that this boss is in a 3d game and doesn't get the ease of design a 2d game boss would get makes it all the more impressive. The RNG is brilliant. The moveset is brilliant. The design of the arena is brilliant. Everything is, I'd nearly assume it was a fluke that Myazaki created a boss this well. 

Sword. Saint. Isshin. Nothing comes even close to how this battle, a 4 phase boss, is put together or is there one I'm overlooking? Videos would be appreciated. 

I've beaten him 4 times (one was a variant of him that was not quite as good) now and gotten to NG+5, each time the RNG keeps it fresh and the increase in difficulty means it's still a slightly greater challenge to complete, staying alive in phase two and three is a feat. I look forward to beating him again all the way to NG+7. A darn masterpiece. 

Hard to really decide which is the "best" for me cause I have a ton of favorites, like I would be listing a ton of bosses for this, so rather im just gonna mention Poseidon from GOW3. Its an introductory boss, but very few games make their introductory boss so epic and I think thats worth mentioning. Not only is a really engaging boss but right off the bat it makes the game's start and presentation instantly memorable. 

Oh, I forgot about God Of War 3. Man, those bosses on the highest difficulty were phenomenal. 

As far as most memorable and intense boss fight would have to be The End from Metal Gear Solid 3. Obviously for different reasons, but talk about an edge of your seat boss fight trying to get to him. I never bothered cheesing him as it was a lot more fun doing it the legit way. Can't wait to do it again in the upcoming remake!

I was thinking Psycho Mantis with his ability to read your memory card and read all of your movements, but once you know switching your controller port, it turns into a pretty easy fight. 

This was still me though

Pyscho mantis was epic the first time when you were clueless, yeah but a gimmick fight is a gimmick fight no matter how cool the gimmick is. I'm excited to see how The End holds up or any MGS3 bosses for that matter, you won't be able to cheese him with modern consoles either, you'll have to legit wait the week to see if it works. 

True King Allant

Pretty epic arena and cool design.

Video for context  

Some of my favourites are DJ Octavio in Splatoon, Raven Beak in Metroid Dread, and Primordial Malzeno in Monster Hunter Rise.

RE4 also has some brilliant boss fights, nearly every one of them a winner.

Vergi DMC 3 and 5.

DMC3. Jester.

Rodin Bayonetta 1/2 secret fight.

Zone of the Enders 2nd Runner. Anubis/Inhert

The final boss Scarlet Nexus is cool.

Jubilee Bayonetta.

Ikaruga. All of them. Seriously amazing.

All of them in Dual Gear on Saturn.

Doponpachi on Saturn same. All of them.

God. SMT III Nocturne.

NMH series has a few I like a lot. Kimmy. NMH3. Shinobu NMH1. Destroy Man NMH2/3. Dr Peace NMH1.

I forget the name but in The Wonderful 101 there is a cool punchout themed fight and the final fights with the Platinum mech are great!

Ys oath in felghana. Galbalan

Red Dragon. Dungeons & Dragons Chronicles of Mystera

Metal Gear Rising. Jetstream Sam. Monsoon. Metal Gear EXCELSUS.

Bogey Alpha and Bogey Bravo. VanQuish.

Nex Machina in Nex Machina. Hard as hell but so so awesome.

Asymptote void in Astebreed

Engels in Nier Automata. YOU ARE FIGHTING THE LEVEL! Beauvoir the opera boss. The Tank. Grun. Emil.

Goda in Yakuza 2.

Stellare Blade. Providence and Raven.

Final Boss Giga Wing 2

Xenoblade X that city sized mech. I forget the name and the final boss was great.

Final boss of Xenoblade 3

Contra III name one.

Hagane. All of them

9 tail Fox in the Otogi games.

Panzer Dragoon has several but Saga has some good ones and Orta might have the best.

I'm missing a bunch from arcades and some side scrolling mech games. Missing a bunch in general but scanning a list of games I own on the shelf these are what I remembered off the top.

Depends what you mean with Best.
If its the hardest I would like to postulate: Cruel Lajos - MOONSCARS: Jesus, It really gave me a headache. In all my years as a gamer (im 38) I never was so close to throw a controller against the wall.
If its the coolest, then: The Boss - MGS3: the way it plays, the music, the difficulty, the significance. Everything in this fight is on point.