Here is the full list of all 32 Exception achievements.
Upgrade a special attack
Complete a level without using all transitions
Collect a byte
Destroy the player
Finish a level with three bonuses
Finish a level with four bonuses
Finish the first level
Achieve a 4-star rating on 32 levels
Achieve a 4-star rating on 64 levels
100% the entire game
Complete the game
Unlock all special attacks
Collect every byte
Destroy 75 enemies with down dash attack
Destroy 100 enemies with sword throw attack
Destroy 50 enemies with up dash attack
Run five kilometers
Jump 2000 times
Destroy 100 turret enemies
Destroy 200 flying enemies
Destroy 150 ground enemies
Fully upgrade a special attack
Fully upgrade all special attacks
Finish 32 levels without destroying an enemy
Finish 64 levels while destroying every enemy
Down dash four enemies in one jump
Up dash four enemies in one jump
Earn an air combo bonus
Earn a style bonus
Escape the monster in world four
Destroy the boss in world eight
Defeat the monster in world twelve