Here is the full list of all 77 Next Hero achievements.
Break more than 100 Objects
First cleared Stage 1
First cleared Stage 2
First cleared Stage 3
First cleared Stage 4
First cleared Stage 5
Kill the King Slime
Kill the King Rabbit Slime
Kill the King Mushroom
Kill the King Lynx
Kill the King Toothtrap
Kill the King Goblin
Kill the Golem
Kill the King Jellyfish
Kill the King Octopus
Kill the King Shark
Kill the King Cobra
Kill the King Toad
Kill the King Turtle
Kill the Siren
Kill the King Ice Slime
Kill the King Bat
Kill the King Kobold
Kill the King Yeti
Kill the King Harpy
Kill the Snow Worm
Kill the Knight Commander
Kill the Horseman Commander
Kill the Chief Ninja
Kill the Chief Archer
Kill the Chief Mage
Kill the Chief Cannoneer
Kill the King Fire Slime
Kill the King Bomb Slime
Kill the King Skull
Kill the King Zombie
Kill the Succubus
Kill the King Salamander
Kill the Dragon
Collect 10 Hearts
Have more than 100 Golds
Have more than 30 Keys
Have more than 30 Bombs
Have more than 10 Equipmets
Drink more than 10 Potions
Open more than 10 Boxes in a game
Enter the Colosseum 3 or more times in a game
First discovered a Secret room
Find a secret room more than 3 times in one game
Death from purchasing items in a Dark shop
Kill more than 100 Monsters
Construction of a mercenary small town
Construction of a blacksmith in a village
Monastery construction in the village
Construction of a pub in a village
Building a museum in town
Achieve highest level of mercenaries
Blacksmiths achieve highest level
Achieve the monastery's highest level
Achieve the bar highest level
Establish a statue of all monsters in the museum
Achieve village level
Marry Sofia
Left the Village
Killed the Elder.
Marrying Diana
Saw the credits.
Completed item collection
Completed monster collection
Acquire all weapons
Acquire all armor
Acquire other items
Acquire all loot
Completed all the quests
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