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