Here is the full list of all 37 Alexis Almighty: Daughter of Hercules achievements.
Complete Chapter One
Complete Chapter Two
Complete Chapter Three
Complete Chapter Four
Defeat all of Chronos' minions
Defeat the Magician of Chronos
Save all the inhabitants in distress
Break all the portals on the first levels
Open all the doors
Use the mana well 50 times
Use the clock 50 times
Save Odysseus
Use the torch 25 times to illuminate the path
Use a skill!
Use a skill 20 times!
Collect 100 food!
Collect 500 food!
Complete Level 1!
Complete 10 levels!
Complete a level with 3 stars!
Complete 10 levels with 3 stars!
Complete 30 levels with 3 stars!
Build 30 buildings!
Set 1000 tasks for workers!
Complete a level without using skills!
Find the first puzzle piece
Find 40 puzzle pieces
Activate the Accelerated Bonus three times at the same level
Use two bonuses simultaneously
Use all the bonuses at least once
Collect 50 materials at any level
Collect 50 food at any level
Complete any level in Gold time without using bonuses
Find 10 puzzle pieces
Find 20 puzzle pieces
Get all the achievements
Complete Chapter Five