SPOILER ALERT: showcasing some of the Memories and Essence that are unlocked by challenge.
Memories

Apply Buffs to allies other than yourself a total of 200 times
Deliver a total of 80 finishing blows to Soul Swordsmen using basic attacks
Deal 1,500 or more damage in a single hit
Take the Merchant’s unattended backpack 12 times (just go back to Merchant when locations occupied by Hunters)
Acquire Unique Memories or Essences from Heroic Bosses 8 times
Fully heal an allied Traveler below 30% Health within 1 seconds
Defeat the Snowy Mountain King while having 4 Cold damage Memories
Enter the Pure White Dream while having 4 memories of the same type
Memories (Mist)

Essence

Use Legendary Chaos a total of 5 times
In a single adventure, visit locations occupied by Hunters 4 times while having a Hunt Threat level of 5 (One of the best essences in my opinion, it stacks maximum health, damage, and healing efficiency)
Apply a total of 3,000 Fire stacks to enemies
Defeat 70 Hunters (Another one of the best essences in my opinion, it stacks AD and AP, and have bonus damage to bosses and hunters)
During a single adventure, let your health fall below 40% and then bring it back up to at least 75% 10 times