Since most fans grew up on the Pokémon franchise, Legendary species have been notably special events in the mainline games. Of course, since the early days of the series with the limitations of technology, several coveted Legendary--namely Mythical--Pokémon were difficult to impossible to come by depending on the part of the world fans were.
Those various Mythical/Legendary events have since somewhat understandably become blander as a sacrifice for much better accessibility as straightforward wireless gifts, but the main and post-game questlines of the games still stand as memorable fan-favorites. As far as the main story goes, this often applied to members of the flagship or Legendary Pokémon trio.
Celebi (Crystal)
The first Legendary event to be a "side quest" rather than the standard on-location trade gift at the time was Celebi in Pokémon Crystal. Though, this event in the early 2000s was no less difficult for players to get their hands on. Only Japan events of different kinds were held to receive one, but through the digital copies from the 3DS Virtual Console port, the event was baked into the game worldwide.
Players would receive a GS Ball from the Goldenrod Pokémon Center (in western versions) to give to Kurt in Azalea Town. As he handles crafting Poké Balls, he analyzes the peculiar item, returning it a day later. After taking it to the shrine in the Ilex Forest, it summons level 30 Celebi for a chance to catch.
Mew (Emerald)
After being a standard gift, one of the best mainline Pokémon games from generation three--Emerald--gave the original Mythical/Legendary Mew a brief side quest. Unfortunately, the key item to find Mew was also region locked in the days of Game Boy Advance. The protagonist would receive an Old Sea Map to be taken to the ship captain in Lilycove City.
After a cutscene, Captain Briney would take them to a remote island in an unknown region called Faraway Island. The mysterious island has tall grass grown throughout, with Mew playfully engaging the player in a puzzle game of hide-and-seek. Once cornering the Pokémon, a battle is engaged for a chance at capture. Mew won't allow cheating, as if the player cuts the tall grass, it flees.
Arceus (Diamond/Pearl/Platinum)
Soon to be getting its own game that could start a series with Legends: Arceus, the Alpha Pokémon was only obtainable in an event quest that was never distributed anywhere--only by cheat codes. To reach Arceus, the Azure Flute was needed to be obtained and taken back to the top of Spear Pillar after beating the Elite Four.
Once the player makes their way back through the Mt. Coronet dungeon, playing the Flute in a certain sequence transports the protagonist to a place identical to Spear Pillar, but with a glass staircase leading to Arceus. The setting, music, and atmosphere made for an exciting encounter and battle with what's the literal Pokémon deity.
Cresselia (Diamond/Pearl/Platinum)
A Legendary event that doesn't require some kind of special item is Cresselia, one half of the Lunar Duo. In Diamond, Pearl, and Platinum after defeating the Elite Four and entering the Hall of Fame, the ship captain in Calalave City, who ran the ferry from there to Iron Island, begs the player to save his son.
He falls into a deep coma-like sleep riddled with nightmares, and the protagonist is taken to Fullmoon Island to find the Lunar Wing that will cure him. There, the player finds Cresselia, who flees but leaves the item behind. Afterward, she'll be a roaming encounter across Sinnoh. It's an interesting side quest to add lore to Cresselia but has larger implications ahead. The lore behind the other half of the duo makes for an eerie connecting of dots.
Darkrai (Diamond/Pearl/Platinum)
The other member of the Lunar Duo is Darkrai, the Pitch-Black Pokémon. It's another exclusive event lost to time but made for a memorable side quest. If the Member Card is received, access is granted to the mysteriously locked inn at Canalave City where the protagonist can spend the night.
But, they're awakened by a nightmare and find themselves on Newmoon Island, where Darkrai is the only resident. There, it can be battled and defeated until captured. For a kid-friendly series, Pokémon games can occasionally make for some entertainingly ethereal, creepy encounters. Fans will likely wonder how event Legendaries like this will play out and be distributed in the upcoming Brilliant Diamond and Shining Pearl.
Victini (Black/White)
A particularly noteworthy Legendary side event was going to catch Victini in the Liberty Garden in the first Black and White games. That's because the event itself incorporated a bit more story than the typical ones. This location can be accessed freely in the pair of sequels, but the Pokémon itself isn't present.
With the Liberty Pass, the protagonist is taken to the Garden, where Team Plasma has taken over with the intention of taking Victini. After battling through five grunts, Victini is in the basement of the building at level 15, and by either defeating or capturing it, Professor Juniper will have enough time to have the Plasma grunts arrested.
Lugia (SoulSilver)
Arguably part of the best Pokémon remakes in fan circles, SoulSilver in particular had a great in-game Legendary event that's part of the mainline quest for the flagship Lugia. It's the first time--and only, so far--Johto has been remade from the original Game Boy Color releases, and seeing the Lugia part of the story felt like a bigger spectacle by Nintendo DS standards.
SoulSilver even tweaked the narrative by adding the Kimono Girls of Ecruteak City to be the ones to summon Lugia in the Whirl Islands. Plus, navigating the Whirl Islands was one of the engaging, fun dungeons and puzzles that the franchise used to have, making the encounter feel more rewarding.
Rayquaza/Deoxys (Omega Ruby/Alpha Sapphire)
The Hoenn remakes became another pair of fan-favorites over the years, and while post-game features like the Battle Frontier and Tower were sidelined, they offered an entertaining post-Elite Four questline. The Delta Episode was a refreshing change story-wise, and put something original into the remakes while incorporating both Rayquaza and Deoxys. It's especially appreciated since Deoxys was one of the Pokémon locked behind bygone events.
A new character, Zinnia, is on a mission to awaken Rayquaza to destroy a meteor. After learning to tame the dragon's power, the player rides it to the meteor, only to have to battle Deoxys as well. This quest proved a welcome addition to post-game content and was a great way to build lore in the world these remakes reconstructed.
Time Travel Celebi (HeartGold/SoulSilver)
It isn't a Legendary event in the traditional sense, but it is one nonetheless. Somewhat similar to the Delta Episode in Omega Ruby and Alpha Sapphire, the promotional Celebi in the Johto remakes would trigger a special story-centric side quest event. When taking the Time Travel Pokémon to the Ilex Shrine in the forest, the player would be taken back in time, revealing that the rival character Silver is the son of Rocket boss Giovanni.
It's shown that Giovanni abandons his son after the latter refuses to join Team Rocket, then going further forward in time for the player to beat Giovanni to help prevent him from reforming his team. This event was another welcome way to inject more story content into the games as well as some interesting new lore.
Giratina (Platinum)
Something many fans likely hope to see a return in the upcoming Sinnoh remakes is the Distortion World from Platinum. Another in-game Legendary event, the climax of the mainline story was the protagonist traveling into the warped world to stop Cyrus and battling Giratina in its Origin Form.
This event was unique, particularly with the technological limitations of the DS, as players needed to incorporate puzzle-solving mechanics along with some light platforming. The game made use of every dimension it could, warping players' perception of direction for a Pokémon game on this handheld, making for a wonderfully trippy experience.