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