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