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