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