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