Here is the full list of all 15 Turing Complete achievements.
Complete the XOR level using only 4 NAND gates
Complete the full adder level using only 5 blue components
Build a Turing Complete computer
Complete the "Logic Engine" level using only components from the "BYTES" category
Obtain the fabled best hat
Hack the hat seller
Complete all 4 quests on the ship
Complete the byte adder with a delay of 40 or less
Complete all the computer architecture levels
Complete the robot race with less than 64 bytes of code
Complete the level called "Conditions" using only 10 blue components
Complete "The Product of Nibbles" using no more than 3 adders
Beat level 6 in the Binary Racing game
Complete the Counter level with a 65 nand score or less
Beat the level "Water World"