Here is the full list of all 16 Hats Are Not Allowed achievements.
Select the Red Hat
Select the Blue Hat
Select the Green Hat
Promoted to C Rank
Promoted to B Rank
Promoted to A Rank
Promoted to S Rank
Select a sword as your weapon
Select the spear as your weapon
Leading from safety or just that good? You sustained zero damage in a battle.
Too easy? You sustained zero damage in a raid.
Defeat General Suppression as the Hatwearer
Defeat Rulerbot without Sustaining Damage
Complete the Revolution Ending
Complete the Power Ending
Complete the Truth Ending