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