Here is the full list of all 54 Mr. Sun's Hatbox achievements.
Retrieve the Hat of Knowledge
Retrieve the Hat of Discovery
Retrieve the Hat of Curiosity
Retrieve the Hat of Power
Retrieve the Hat of Judgment
Return Mr. Sun's Hatbox
Completely fill the base with rooms
Complete a mission without killing anyone or setting off an alarm
Capture 100 items
Recruit 100 units to join your staff
Successfully complete an emergency mission
Escape a level inside a frog
Capture a mimic
Capture a flying enemy
Complete all your daily contracts for the day
Upgrade 20 different weapons in the armory
Reach Level 10 with a staff member
Reach Level 20 with a staff member
Sell 1000 gold worth of items from the Sales room
Gain 1000 or more XP from completing a single mission
Complete a Very High difficulty mission using a unit with 6 quirks that are all negative or inherent (without taking backups)
Lose a staff member that is level 10 or higher
Complete a Very High difficulty mission using a unit with 6 quirks that are all neutral or inherent (without taking backups)
Complete a Very High difficulty mission using a unit with 6 quirks that are all positive or inherent (without taking backups)
Bring 5 or more backups into a mission... and lose them all
Kill 3 or more enemies with a single slam of a Slammer hat
Die shortly after slipping on something
Die after successfully completing a mission
Capture an enemy inside a soup pot
Escape a cloud level via parachute hat
Snap a Knight's neck while they're still fully armored
Complete the final mission without taking backups
Complete a Spec Ops mission with 15 or more squad members
Research everything
Extract 50 quirks in the lab
Recruit every type of unit
Obtain 40 different kinds of hats
Obtain 30 different kinds of weapons
Bring a blindfold into a mission and complete it without taking it off
Complete a raid without directly killing anyone
Complete a Very High difficulty mission with a staff member who has Spindly Arms, Weak Throw, and Lightweight
Kill an enemy by reflecting a bullet back at them
Capture a gold knight
Complete 50 missions with a single unit
Turn 5 enemies into frogs on a single floor
Capture a glass knight
Capture or kill an enemy who killed one of your staff members
Rescue a staff member that you previously abandoned
Watch someone die from being hit by their own boomerang
Gather 1000 or more gold in a single mission
Snap a skeleton's neck as a skeleton while wearing a skull on your head
Sell Mr. Moon's Organs
Complete a Very High difficulty mission while listening to the Circa Infinity soundtrack the entire time
Snap a picture of Mr. Moon