Here is the full list of all 9 Cake Game achievements.
The base game contains 6 achievements, and there is 1 DLC pack containing 3 achievements.
Get the mama cake on easy difficulty
Get the papa cake on easy difficulty
Get the mama cake on medium difficulty
Get the papa cake on medium difficulty
Get the mama cake on hard difficulty
Get the papa cake on hard difficulty
Get two papa cakes on easy difficulty
Get two papa cakes on medium difficulty
Get two papa cakes on hard difficulty