The following are all rewards obtained by completing each challenge. All of them are recipes that can only be used in Terra Incognita (Unlocked by beating the first chapter for the first time). Access to the Challenge progression is also unlocked for each chapter you have beaten. Which can then be viewed from the main menu screen > Chapter of Choice > View Challenges (Press Square when selecting New/Load Game).

Reward	How to unlock
[Item/Equipment] Display Stands	Chapter 1: Defeat the three Dragons.
Costume Set	Chapter 1: Build the Cantlin Garden.
Erdrick's Legendary Items and Gear	Final Chapter: Complete the chapter within 30 days.
Featherwear Footfall Accessory (No Fall Damage)	Chapter 1: Repair the Hammerhood's Gravegard.
Flame Blade	Chapter 1: Complete the chapter within 20 days.
Forbidden Furniture	Chapter 3: Defeat all the Boss Trolls and Gigantes.
Forbidden Gear	Final Chapter: Find the Sword of Ruin.
Gourmand's Girdle Accessory (Hunger stops)	Chapter 2: Complete the chapter within 30 days.
Hot Water Crystal	Chapter 3: Complete the Spectacular Spa blueprint.
Lyres of Slime Immemorial (For all modes, not just Terra Incognita; Changes all BMGs to 8-bit Chiptune when placed)	Chapter 3: Obtain recipe for the lyre of slime immemorial.
Meteorite Bracer Accessory (Run Faster)	Chapter 2: Complete all of Thalamus's puzzles.
Novelty Items	Final Chapter: Show a verdant vision to a certain someone.
Ring of Criticality Accessory	Final Chapter: Defeat the Final Boss without wearing the legendary armor.
Roofing Tiles set	Chapter 2: Repair the ruined roof.
Special Blocks (Hardwood and Timber)	Chapter 3: Build your Base to Level 5.
Stone Furniture	Chapter 1: Build your Base to Level 5.
Superior Crafting Station (everything can be built here)	Final Chapter: Build your Base to Level 4.
Talaria Accessory (Double Jump)	Chapter 3: Complete the chapter within 30 days.
Water Crystal	Chapter 2: Obtain the Crown Goowels.
Wood Furniture	Chapter 2: Build your Base to Level 5.