Here is the full list of all 26 Invasher achievements.
Defeat boss in stage 1
Defeat boss in stage 2
Defeat boss in stage 3
Defeat boss in stage 4
Defeat boss in stage 5
Defeat boss in stage 6
Defeat boss in stage 7
Defeat boss in stage 8
Defeat boss in stage 9
Finished game in main game mode
Finished game in main game mode under 30 minutes
Finished game in main game mode under 45 minutes
Finished game in main game mode under 60 minutes
Finished game in main game mode with no death
Defeat boss in stage 10
Defeat boss in stage 11
Defeat boss in stage 12
Defeat boss in stage 13
Defeat boss in stage 14
Defeat boss in stage 15
Reached 9999 GP
Finished the game without buy any weapon
Finished the game with 5 max HP and max EP
Owned all weapons
Upgraded HP and EP to maximum
Collected all golden eggs