Here is the full list of all 17 Stay Close achievements.
Don't get caught by a Creeper
Finish the Game in 30 Minutes
Finish the Game without dying
Win one SP game
Win 10 SP games
Win one Coop Game
Win 10 Coop Games
Read all notes
Find Emperor Olaf
Find the supporter wall
Find Cyrus
Find them
Find the key
Fulfill Cyrus' game
Find the compass
Find Jeeny
Find Maya