Here is the full list of all 37 The Slimeking's Tower achievements.
Mind Completed
Mind Unlocked
Prayer Completed
Dots Completed
Prayer Unlocked
Dots Unlocked
Hairry Completed
Jade Completed
Igneous Completed
Maiyan Completed
Midas Completed
Void Completed
The Collector Completed
Beat the portal
Reach The Portal
Midas Unlocked
Void Unlocked
Reach Sigma
Reach Theta
Reach Tau
Reach Ohm
Reach Pi
Defeat Lodeslime
Reach the basement
Jade Unlocked
Reach Omega
Reach Gamma
Reach Epsilon
Reach Delta
Igneous Unlocked
The Collector Unlocked
Maiyan Unlocked
Attack a mimic before it's transformed
Reach Beta
Get your first boss key
Play Adventure Mode for the first time
Unlock Hairry's secret item