Here is the full list of all 27 Great Hero's Beard achievements.
Max out all princesses
Defeat the Chubby Tubby Warlord
Defeat the Sea King Neposeitune
Defeat the Avalanche
Defeat Skelethrone King Slayer
200 ATK SPD
Defeat Demon Lord Baal
100'000 HP
10th Retirement
Defeat the Guardian Azriel
Rescue Fifth Princess
5th Retirement
First Retirement
Reach Level 100
Rescue 4th Princess
Reach Level 80
100'000 Damage
Deal 1M Damage
Rescue Third Princess
Reach Level 60
100 ATK SPD
Rescue Second Princess
Deal 1'000 Damage
Reach Level 40
Rescue First Princess
Reach Level 20
First Fusion