Here is the full list of all 38 Potion Permit achievements.
Unlock all achievements
Pet your dog 50 times
Forage 1,000 stone
Forage 1,000 tree
Work part-time in the Church 15 times
Forage 1,000 plants
Defeat 1,000 enemies
Have a bath in Willow Waters Bathhouse 10 times
Cook food 50 times
Cook food 15 times
Collect 10 pieces of furniture
Date an NPC for the first time
Failed to cure 10 patients
Failed to cure patient for the first time
Work part-time in the Post Office 15 times
Clear 75 quests from the Community Board
Work part-time in the Police Department 15 times
Successfully cure 50 patients
Cook food 5 times
Brew potions 150 times
Collect 10,000 stone
Reached Friendship Lv.3 with a villager
Reach Friendship Lv.2 with 10 villagers
Clear 25 quests from the Community Board
Collect 10 type of fish in Moonbury
Collect 3,000 wood
Brew potions 50 times
Collect 10,000 coins
Roll for 1,000 times
Successfully cure 10 patients
Go fishing 50 times
Brew potions 15 times
Reached Friendship Lv.1 with your dog
Reach Friendship Lv.1 with 5 villagers
Passed out for the first time
Use fast travel 20 times
Brew your first potion
Start a new game