	Spoiler warning: This article may contain major plot or ending details.

This guide lists all of the events you can complete after beating the Pokémon League. The information was taken from this forum post.

Quests

The Timeless

After you beat the E4, go back to the Crystal Caves south of Amphitrite City where you battled Mew where the Timeless are waiting.
Defeat all of them and battle their leader to become the 3rd Augur of the Torren Region and unlock other post game quests.

Recommended Levels: 85-95

The Holon Region

After beating the Timeless you will get a call from Scientists Club if you already finished the mission. They ask you you to go to Suntouched City’s Subway.
The subway will take you to the Holon Region where you can obtain several Delta Pokemon and the Regi Trio (and their Delta counterparts).
While going through the Holon Pyramid, you’ll find the Magic Carpet, the replacement for HM Waterfall.

Recommended Levels: 90-100

Taen's Quest

After getting a call from Orion, you head to his gym where he briefs you on his situation. He tells you that Taen is back in town and is gonna try to summon the Original Dragon once again.
Head to the Perfection Base where you find Reukra being berated by angry protesters including one green haired individual named Gail. He shows you his Zygard in a friendly spar and gives you a pair of Hiking Boots, the replacement for HM Rock Climb.
From here, go to Gaea Town and use the Hyperspace Hole inside the old gym. You will be transported to No Man's Land.
Head north and enter Alpha Temple. At the end of the temple you will encounter Taen; beat him and heal at any Pokemon Center.
From here, go to the far west wing in Utira Library to find Orion. Listen to their conversation and then return to Dragon Ruins.
After your fight with Taen, Zygard will start rampaging. You can catch it and then collect the stones for Zekrom and Kyurem.
Then Taen leaves the region to star in Pokemon Azurite, marking the end of Taen's Quest.

NO DATA (optional)

Find the Human Calculator in Gaea Town, tell him to divide by 0 and watch your game crash.
When you reopen the game, hop onto the EmolgaTaxi and soar to the southeast of Torren. You will find a buggy mess in the ocean, which say’s it’s “Gaea Town”.
Head inside and battle MISSINGNO.

Tip: Hitting it with physical attacks will have a chance to instantly faint your Pokemon. After finishing the battle with it, you will be sent back to Gaea Town.

Recommended Levels: 85-90

Relic Song (optional)

Find all 8 Mysterious Scrolls (guide to completing the puzzles).
Go to northeast Samsara Cave entrance in Route 5.
Follow the paths and continue exploring the cave, releasing locks using the Mysterious Scrolls in the process.
At the end, battle the level 100 legendary Meloetta and obtain the Relic Song ability.

Recommended Levels: 85-90

Damian's Quest

After obtaining the Hiking Boots (Rock Climb) and the Magic Carpet (Waterfall), go to the middle cave entrance in Route 5.
Once the player has obtained the ability to use Waterfall and Rock Climb to return to the Dream Realm, the player is immediately thrown into a conversation with Adam in his Gym shortly after reentering Erebus City.
After the conversation with Adam finishes, the player is warped to the Prion Site and is told to go west into the Nightmare Badlands.
As you climb through the Badlands, battling as you go, eventually you will get to a little shack, before which a rather powerful trainer will accost and battle you.
After you win, enter the shack to catch Darkrai.

NOTE: Remember where that shack is. you will want to go back there eventually in Quest 9: Diana’s Parcel.

Recommended Levels: 90-95

The Azure Flute (optional)

After finishing the Original Dragon, Holon, and Dream World quests, go back to Perfection Base (in the Black Market).
Once you read the note, return to the Crystal Caves to the cave where Mew and the Timeless were.
Enter the portal which you previously used to get to Giratina and follow the path the same way you did for Giratina.
Eventually, you will get back to the Throne of the Hegemon, and Reukra will battle you
After the battle Arceus will be summoned which you can catch (NOTE: Arceus has Perish Song, so be very careful).

NOTE: This quest must be finished before the Pokemon League can be re-matched.

Recommended Levels: 100-110.

Rose Crater Quest (optional)

After beating the Timeless, go to Deyraan Town. Head for Rose Crater and you will see professor’s aid, who dropped the magma stone in Infernal Base.
Go to Infernal Base, navigate through and you will see Heatran and Volcanion fighting. Navigate until you find a path blocked by magma, then equip the magma armor and go through the magma.
After some navigation, you will find Groudon. Fight it and beat/catch it.
Then head towards the exit, and you will encounter Heatran and Volcanion in a double battle, where you can catch one of them.
Exit the base and talk to the aid for reward.

NOTE: If you catch Heatran and kill Volcanion, Volcanion will appear in the Cave of Steam. If you catch Volcanion and kill Heatran, it will appear in Fiery Caverns.

Recommended Levels: 90-95

Diana's Parcel (optional)

Requirements: Hiking Boots, Magic Carpet, Hyperspace Hole.

After finishing the Azure Flute Quest, go to the Selene City Gym and talk to Diana and she will give you a Parcel.
Go to Route 5, then the Dream Realm.
Go to the Rem Village where you caught Darkrai and talk to Persephone. She will take the parcel and give you a Master ball. This also allows you to be able to catch Cresselia as a roaming legend.
Go back to Selene City and talk to Diana.

Recommended Levels: 90-100

Audrey’s fate

After healing Damian, he calls you and tells you to go to Maelstrom 9. When you get there it is raining heavily.
Dive down and go to the Abyssal Base. Make your way to the end where you previously fought Audrey.
Dive down further and follow the path, which is dark in some spots so the Flashlight may be useful.
Eventually you will meet Damian, and see a battle between Lugia and Kyogre, with Lugia trying to protect Audrey. While Kyogre is distracted Damian uses an escape rope to rescue his mom, which directs both legendary pokemons’ attentions towards you.
You then battle alongside Lugia against Kyogre and a Tentacruel. You can catch Kyogre in this battle and Lugia tries to aid this by using Thunder Wave.

Recommended Levels: 95-100

Perfection Base

After finishing the Azure flute quest, go back to Perfection Base. The police are all over the base. (You can read the diaries in the previously locked room to find a key and a Delta Volcarona armor).
There is also another passageway opened by the police behind where a computer used to be. Inside is a series of boulder-pushing puzzles (if needed you can go back up the stairs to reset the puzzle).
Once the puzzle is solved Mewtwo will appear and be catchable.

Perfection Laboratory

After obtaining the Laboratory Key in the Perfection Base, go to Metchi Town and surf to the north.
Use the key to enter the Perfection Laboratory. Use Seed Flare to go further into the lab.
Enter the room above and watch the security tape on the top right computer to find a code. Use this code on the top right side of the lab to access a Heart Swap statue that can be placed on a pink/red switch in the middle.
Go to the bottom right to find a now-accessible staircase. Check the bookshelves to find clues to the password, which is “antilithium”.
Enter the password to unlock the door to Delta Larvesta in the top left. Don't forget the Mewtwo Armor on the left.