Yes, we all know that beating a Gym Leader is awesome, but to get real satisfaction out of the ordeal, try beating them with underleveled Pokémon. Or take it up a level and do the same thing with the Elite Four and Champion.
The Generation V games brought it up a little when the music shifts to an upbeat remix of the classic Pokémon theme when the leader is on his/her last mon.
Mewtwo. Generation I. You've gone through the Pokémon League, tackled Articuno, Zapdos, and Moltres, proved once and for all that you're better than Blue, and now you're at the end of Cerulean Cave. There you see it before you: a single solitary Pokémon. You remember the journal entries from Cinnabar Mansion: the scientists finding Mew and cloning it to make Mewtwo, who proceeded to escape and destroy the laboratory/mansion. This is Mewtwo. Level 70. And you can catch it. The only way to ruin that fight is to use the Master Ball, in which case it's an Anticlimax Boss.
Another way to make it an anticlimax boss is to fight him with a Dark-type in FRLG/HGSS, allowing you to catch him with baseline Poké Balls with total ease, if a bit of a time investment. As Dark-types are unaffected by Mewtwo's Psychic-type offensive moves, almost the only thing you have to worry about is Mewtwo running out of PP.
Then there's the solo battle in Pokémon Stadium. That he challenges you is his way of saying, "Hey. I can beat you and your whole team by myself. Bring it on!" The arena is made of a bunch of transparent crystalline material, and he has music befitting a final boss. And because the balancing issues hadn't been sorted out yet, he stood a pretty good chance of actually wrecking your team all by himself.
The True Final Boss of the second generation and its fourth-generation remakes. You've obtained sixteen badges, and defeated every Gym Leader within your reach. Your team of mons is fully-evolved and totally ready and pumped to kick some ass. You've literally defeated every other trainer in the two regions. Who could possibly be left to challenge you? Who else butRed, the protagonist of the previous game, who after five generations still wields the highest-leveled Pokémon in the series. note His HGSS and GSC teams are the first and fourth-highest leveled trainers respectively, excluding Battle Frontier facilities but including Benga's Challenge Mode team from BW2, who is tied for second with Barry from Platinum. He's got some of the most famous and traditional Pokémon: the three fully evolved Kanto starters, Snorlax, Pikachu, and either Espeon or Lapras depending on version. As if all that wasn't enough, he doesn't even '' say'' anything as you fight him, speaking only in ellipses as a reference to his own silence in the original games. It was an incredible way to tie the first two generations together and bring everything full circle. It speaks volumes of how awesome this fight is when there is absolutely no dialogue or context involved and it still manages to feel epic based solely on the Previous Player-Character Cameo.
Tate and Liza in Ruby/Sapphire/Emerald. They're the only double-battle Gym Leaders in the main games (rematches in Emerald notwithstanding) and require the player to push their double-battle strategies to their limits—or to come with one on the spot—if they haven't been Level Grinding. The battle may take longer than most battles up to that point (even though there are only two Pokémon (four in Emerald), but that's part of the charm of RPGs bosses for some players.
Steven in Ruby/Sapphire (he apparently abdicated his Champion position to Wallace in Emerald, but he remained in the game as an even stronger Bonus Boss). Although he was the first Final Boss in the series to not have a direct connection to the player (as opposed to your rival in the first generation and the below-mentioned Red in generations two and four), he overcame that disappointment with an incredibly difficult-to-defeat team and a kickass battle theme. He also had a pretty cool, easygoing personality... and an interior as hard as the Steel-types he uses. Pretty badass.
Some of the later battles against Norman in Emerald can feel like this. It's not just the thrill you get after taking out his skill-swapping Slaking, which received a negative ability just to keep them balanced. It's what he says after you beat him: "<Your Name Here>... What is your dream? My dream... Hahaha...It has already come true, actually." To summarize: it's not just beating him, it's that instead of resenting that his child has defeated him multiple times, he's genuinely happy for you.
The Sinnoh Elite Four. Hell, the battle with Cynthia alone. (You can't exploit type weaknesses against her Spiritomb, because it has no weakness.)
Cynthia is generally improbably cool. Not only she has a kickass well-trained team, an awesome battle track, but she is an appealing character as well, walking around and helping the player character because she thinks newbies are cool.
Cynthia was so cool they added her in as an optional battle in Black and White and Black 2 and White 2.
The Giratina fight following it also is very awesome, due no doubt to the music and just the fact that you're fighting a powerful being that probably isn't too happy with what Cyrus was doing and probably thinks you're part of it. It's also a pretty tough fighter in its own right and won't give up easily.
N and Ghetsis from Black and White play things a lot more differently than the villains in the other main Pokémon games. They are fought RIGHT after defeating the Elite Four. You first fight the mascot legendary - which respawns if it faints, forcing you to catch it, then you fight N, who has the other version's mascot legendary as part of his team, then you fight six more Pokémon from Ghetsis.
Add the fact that he is in a very short list of villains who have actually obtained a Legendary Pokémon and use it against you. The only other ones to do so? Cipher.
Ghetsis and more specifically, his HYDREIGON. That meta-gaming jerk has a mid-50s Hydreigon (the game mechanics have it evolving at 64), plus it's loaded down with SURF, FIRE BLAST, FOCUS BLAST, and... Dragon Pulse?... Anyway, so yeah. It'll just about oneshot ANYTHING it can line up a type weakness against. So your list of types to fight it with are slim. Best option is a PURE Fighting-type, as it is a Dark / Dragon. Barring that, a Water-type with Ice moves (Samurott and/or Jellicent) works, too.
In Black 2 and White 2, Ghetsis is back. His Hydreigon was dumbed down quite a bitnote its moveset is physical-based instead of special-based and is walled by Steel-types, not to mention you have plenty of other options to combat it with thanks to the New Unova Dex, and two of his former Pokemon (Bouffalant and Bisharp) were replaced with Poison-types from Sinnoh, but it still makes for a cool fight. His theme was also revamped pretty hardcore, too.
On a similar note, Colress from Pokémon Black 2 and White 2, who you fight before Ghetsis. Despite four of his five Pokémon being Steel-type (six in post-game rematches), his Magneton and Magnezone can really mess with you if you lack an answer to them. His other Pokémon can hit hard as well, but he does make for a fun battle, especially with his technologic and intense battle theme.
Battling the Champion, Iris, in Pokémon Black 2 and White 2. After an outstanding battle intro scene, which is the only one the series has had yet to feature full body VS sprites, the battlefield is a bunch of multicolored beams flying through the starry sky. You know you're in for a fun fight when the Champion's first Pokémon is a Hydreigon. Anotherunderleveled one, no less, which, in fact, if you're playing on Challenge Mode or a rematch, has the same moveset that Ghetsis' had in the originals note In a regular initial battle, it has Flamethrower and Charge Beam instead of Focus Blast and Fire Blast. And that's just the beginning. She also has a Haxorus with Dragon Dance and a Focus Sash, along with lots of other strong hitters. While she's not the hardest Champion battle you'll face, it still makes for an awesome fight, especially with the pure undiluted awesome that is her battle theme. And on Challenge Mode it's even more intense, because her entire team gets redesigned and are given much more powerful movesets consisting of 100-120 and even 150 power STAB moves.
Additionally, on Challenge Mode, she and the entire Elite Four can be considered this, because in addition to them having five Pokémon like they did in previous games prior to Black and White, they all get unusual held items that boost the power of their attacks or accuracy (Wise Glasses, Expert Belt, etc). This one takes an all-out offensive force to be reckoned with.
Elesa's revamped gym in Black 2 and White 2. It consists of a gaudy catwalk where you fight supermodels, who add to the funky-fresh mix of the normal gym music when you defeat them (complete with Gratuitous English!). It's more of a funny boss than an awesome boss, but it's still plenty great.
Benga, Alder's grandson also from Pokémon Black 2 and White 2 is a pretty awesome fight as well. You battle him at the top of the Black Tower/White Treehollow which gives the battle with him a unique background. Getting to him is a long and arduous trek through extremely high leveled trainers with very limited chances to heal. When you finally do reach him, he comes flying at you with three extremely powerful Pokemon with good hold items that are even higher leveled than Iris'. One of them (depending on the version) is even a Latios or Latias! Finally, during the battle, him and his grandpa's heroic and triumphant theme is blasting in the background, giving the fight an incredible feeling of grandeur.
The climax of the Team Flare arc. First, the mascot legendary battles. As with Zekrom and Reshiram, it respawns if it faints and forces you to catch it, but in the middle of doing so, you get thrown into an outstanding battle scene, with Xerneas in what looks like a rainbow-ish field, and Yveltal in outer space, with their absolutely mind blowingly awesome battle theme to throw on it. Then you fight Team Flare's boss, Lysandre, for the final time, with his Doc Ock-style robotic limbs on his back, on the sun. He's no BW Ghetsis in terms of difficulty, but it's still pretty sick, not to mention you can use your newly acquired legendary to kick his butt.
Yeesh, is this guy a joke? Mega Blaziken took all of them down with ease! Oh, wait, a Gyarados? Ah, no worr- The foe Gyarados's Gyaradosinite is reacting to Lysandre's Mega Gauntlet! And you thought only you could use Mega Evolution. NOPE!
Also in PMD2, battles with Dusknoir seem to be magnets for Awesome Moments.... that is, if you're not busy having a Heroic BSOD from the plot developments.
Explorers of Sky adds to this with the final secret mission. Grovyle, Dusknoir, and Celebi versus Primal Dialga as they all slowly disappear due to the timestream repairing itself. It ends with all of them being revived by Arceus and getting to see the first sunrise of their lives. A perfect end to Grovyle's story.
In Guardian Signs, there's the fight against Purple Eyes, who is using the full power of the Golden Armour. You just captured Ed's Mewtwo in a fight that was both extremely tough and seriously awesome, but now you have to do it again with Mewtwo surrounded by an aura of dark energy, which not only increases the power of Mewtwo's attacks, but also counts as an attack in its own right. This is way harder and considerably more awesome.
XD001. Becomes an Anticlimax Boss with the Master Ball, and even without the Master Ball it's easy to KO. Just not to catch.
If you want an awesome moment, in Colosseum try Evice. Not just for the amazing music, but for the fact that when you work out just how well your key Pokémon fit against his ( Entei beats Scizor, Suicune beats Salamence, Raikou beats Slowking, Espeon beats Machamp, Umbreon walls Slaking to death) you realise you've truly mastered the game.