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
10th Retirement
200 ATK SPD
Defeat Demon Lord Baal
100'000 HP
Defeat the Guardian Azriel
5th Retirement
Rescue Fifth Princess
First Retirement
Reach Level 100
Rescue 4th Princess
Reach Level 80
Deal 1M Damage
100'000 Damage
100 ATK SPD
Rescue Third Princess
Reach Level 60
Deal 1'000 Damage
Rescue Second Princess
Reach Level 40
Rescue First Princess
Reach Level 20
First Fusion