Here is the full list of all 28 Battle Axe achievements.
DESTROY 25 GENERATORS
SLAY 500 ENEMIES
DESTROY A CHAIN OF 100 ENEMIES
DESTROY 10 GENERATORS
SLAY 250 ENEMIES
SLAY 100 ENEMIES
DESTROY A CHAIN OF 50 ENEMIES
KILL 5 ENEMIES IN ONE GO
SLAY 50 ENEMIES
DESTROY A CHAIN OF 10 ENEMIES
ACHIEVE AN 'S' RANK ON AREA 1
ACHIEVE AN 'S' RANK ON AREA 2
ACHIEVE AN 'S' RANK ON AREA 3
ACHIEVE AN 'S' RANK ON AREA 4
DESTROY A CHAIN OF 250 ENEMIES
DESTROY 50 GENERATORS
RESCUE 50 VILLAGERS
RESCUE 100 VILLAGERS
COMPLETE LEVEL 10 OF INFINITE MODE
COMPLETE LEVEL 25 OF INFINITE MODE
COMPLETE LEVEL 50 OF INFINITE MODE
AMASS 1000 COINS
COMPLETE THE GAME ON HARD
COMPLETE NEW GAME+
COMPLETE GAME WITHOUT BUYING ANYTHING
COMPLETE AN AREA WITHOUT MISSING A PROJECTILE ATTACK
SCORE ONE MILLION POINTS
COMPLETE GAME WITHOUT DYING