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