Here is the full list of all 28 Hegemony III: Clash of the Ancients achievements.
Maintain military hegemony for 2 weeks
Win a sandbox as a Greek faction
Maintain trade hegemony for 2 weeks
Harvest 100 000 wood
Defeat 500 Brigades
Maintain cultural hegmony for 2 weeks
Have a total population of at least 100 in a single game
Capture 1000 slaves
Defeat 100 Brigades
Win a sandbox as a Latin faction
Win a sandbox as an Etruscan faction
Win a sandbox as a Gallic faction
Win a sandbox as a Sabellic faction
Win a sandbox as an Illyric faction
Conquer all cities in the Unification sandbox
Maintain naval hegemony for 2 weeks
Defeat 1000 Brigades
Harvest 100 000 000 food
Mine, tax or loot 100 000 gold
Raid 100 resource buildings from 0 to 100
Raze away 100 points of city population
Get the Tarentines to accept Pyrrhus' control of their city
Defeat Valerius Laevinus
As Pyrrhus, defeat the Roman army gathering at Asculum
As Pyrrhus, protect Syracuse from Carthaginian invasion
Proclaim Pyrrhus as King of Sicily by taking Eryx and Lilybaeum from Carthage
Win a sandbox as a Sicilian faction
Control every city of Sicily