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Refractal Function
The Refractal Function (RFF) is not a fast growing function but I thought it was cool and posted it here.

It is defined as: Here are the decompositions of RFF(x) from 0 to 6. Solving of RFF(0) {0,0} {0} RFF(0) = 2 Solving of RFF(1) {1,1} {0} RFF(1) = 2 Solving of RFF(2) {2,2} {1,1} {0} RFF(2) = 3 Solving of RFF(3) {3,3} {2,2,2} {2,1,1} {2,0} {2} {1} {0} RFF(3) = 7 ​​​​​​​Solving of RFF(4) {4,4} {3,3,3,3} {3,3,2,2,2} {3,3,2,1,1} {3,3,2,0} {3,3,2} {3,2,2} {3,1,1} {3,0} {3} {2} {1} {0} RFF(4) = 12 ​​​​​​​Solving of RFF(5) {5,5} {4,4,4,4,4} {4,4,4,3,3,3,3} {4,4,4,3,3,2,2,2} {4,4,4,3,3,2,1,1} {4,4,4,3,3,2,0} {4,4,4,3,3,2} {4,4,4,3,2,2} {4,4,4,3,1,1} {4,4,4,3,0} {4,4,4,3} {4,4,3,3,3} {4,4,3,2,2,2} {4,4,3,2,1,1} {4,4,3,2,0} {4,4,3,2} {4,4,2,2} {4,4,1,1} {4,4,0} {4,4} {3,3,3,3} {3,3,2,2,2} {3,3,2,1,1} {3,3,2,0} {3,3,2} {3,2,2} {3,1,1} {3,0} {3} {2} {1} {0} RFF(5) = 31 ​​​​​​​Solving of RFF(6) {6,6} {5,5,5,5,5,5} {5,5,5,5,4,4,4,4,4} {5,5,5,5,4,4,4,3,3,3,3} {5,5,5,5,4,4,4,3,3,2,2,2} {5,5,5,5,4,4,4,3,3,2,1,1} {5,5,5,5,4,4,4,3,3,2,0} {5,5,5,5,4,4,4,3,3,2} {5,5,5,5,4,4,4,3,2,2} {5,5,5,5,4,4,4,3,1,1} {5,5,5,5,4,4,4,3,0} {5,5,5,5,4,4,4,3} {5,5,5,5,4,4,3,3,3} {5,5,5,5,4,4,3,2,2,2} {5,5,5,5,4,4,3,2,1,1} {5,5,5,5,4,4,3,2,0} {5,5,5,5,4,4,3,2} {5,5,5,5,4,4,2,2} {5,5,5,5,4,4,1,1} {5,5,5,5,4,4,0} {5,5,5,5,4,4} {5,5,5,5,3,3,3,3} {5,5,5,5,3,3,2,2,2} {5,5,5,5,3,3,2,1,1} {5,5,5,5,3,3,2,0} {5,5,5,5,3,3,2} {5,5,5,5,3,2,2} {5,5,5,5,3,1,1} {5,5,5,5,3,0} {5,5,5,5,3} {5,5,5,4,4,4} {5,5,5,4,3,3,3,3} {5,5,5,4,3,3,2,2,2} {5,5,5,4,3,3,2,1,1} {5,5,5,4,3,3,2,0} {5,5,5,4,3,3,2} {5,5,5,4,3,2,2} {5,5,5,4,3,1,1} {5,5,5,4,3,0} {5,5,5,4,3} {5,5,5,3,3,3} {5,5,5,3,2,2,2} {5,5,5,3,2,1,1} {5,5,5,3,2,0} {5,5,5,3,2} {5,5,5,2,2} {5,5,5,1,1} {5,5,5,0} {5,5,5} {5,4,4,4,4,4} {5,4,4,4,3,3,3,3} {5,4,4,4,3,3,2,2,2} {5,4,4,4,3,3,2,1,1} {5,4,4,4,3,3,2,0} {5,4,4,4,3,3,2} {5,4,4,4,3,2,2} {5,4,4,4,3,1,1} {5,4,4,4,3,0} {5,4,4,4,3} {5,4,4,3,3,3} {5,4,4,3,2,2,2} {5,4,4,3,2,1,1} {5,4,4,3,2,0} {5,4,4,3,2} {5,4,4,2,2} {5,4,4,1,1} {5,4,4,0} {5,4,4} {5,3,3,3,3} {5,3,3,2,2,2} {5,3,3,2,1,1} {5,3,3,2,0} {5,3,3,2} {5,3,2,2} {5,3,1,1} {5,3,0} {5,3} {4,4,4,4} {4,4,3,3,3,3} {4,4,3,3,2,2,2} {4,4,3,3,2,1,1} {4,4,3,3,2,0} {4,4,3,3,2} {4,4,3,2,2} {4,4,3,1,1} {4,4,3,0} {4,4,3} {4,3,3,3} {4,3,2,2,2} {4,3,2,1,1} {4,3,2,0} {4,3,2} {4,2,2} {4,1,1} {4,0} {4} {3} {2} {1} {0} RFF(6) = 99 Notice the recursive patterns decomposing it makes.
 * 1) Make an array with 2 numbers, both of them the number you put in the function.
 * 2) Recursively apply the following until the array only has a zero:
 * 3) If the number at the end of the array is 0, remove it.
 * 4) Otherwise, replace the last two entries in the array with B copies of A.
 * 5) A is the second-to-last entry in the array, and B is the last entry in the array.
 * 6) If there is only one entry in the array, decrease it by one.
 * RFF(0) = 2
 * RFF(1) = 2
 * RFF(2) = 3
 * RFF(3) = 7
 * RFF(4) = 12
 * RFF(5) = 31
 * RFF(6) = 99