#Br**nfuck, 128-5=123 bytes

Br**nfuck, 128-5=123 bytes

Cell 3 (0-indexed) is the running total x. +<<+<<,<, Initialize tape with [n, a, 0, 1, 0, 1] [ While n: >[>+>>+<<<-]>[<+>-]  Copy a 3 cells to right: [n, a, 0, x, a, 1] >[ While x: >[>>+>+<<<-]>>>[<<<+>>>-]  Copy a 2 cells to right: [n, a, 0, x, a, 1, a, 0] <<[>[>+>+<<-]>>[<<+>>-]<<<-]  Cell 7 = prod(cell 5, cell 6) >[-]>[<<+>>-]<<<<-]  Move this value to cell 5. End while. >>[<<+>>-]+<[-]<<<<-]  Update x to result of exponentiation. End while. >>>. Print the result! 

This works (tested) for x 0, 0 x, x 1, 1 x, x 2, 2 3, and 2 4. I tried 3 3, but it ran for several hours without finishing (in my Java implementation—probably not optimal). In theory, this works up to the cell size of the specific implementation.

Cell 3 (0-indexed) is the running total x. This calculates the nth tetration of a.  +<<+<<,<, Initialize tape with [n, a, 0, 1, 0, 1] [ While n:  >[>+>>+<<<-]>[<+>-] Copy a 3 cells to right: [n, a, 0, x, a, 1] >[ While x:  >[>>+>+<<<-]>>>[<<<+>>>-] Copy a 2 cells to right: [n, a, 0, x, a, 1, a, 0]  <<[>[>+>+<<-]>>[<<+>>-]<<<-] Cell 7 = prod(cell 5, cell 6)  >[-]>[<<+>>-]<<<<-] Move this value to cell 5. End while.  >>[<<+>>-]+<[-]<<<<-] Update x to result of exponentiation. End while. >>>. Print the result! 

This works (tested) for x 0, 0 x, x 1, 1 x, x 2, 2 3, and 2 4. I tried 3 3, but it ran for several hours without finishing (in my Java implementation—probably not optimal) (EDIT: in @Timwi's EsotericIDE [It's great! Y'all should try it] as well. No luck.). In theory, this works up to the cell size of the specific implementation.

#Br**nfuck, 128-5=123 bytes

+<<+<<,<,[>[>+>>+<<<-]>[<+>-]>[>[>>+>+<<<-]>>>[<<<+>>>-]<<[>[>+>+<<-]>>[<<+>>-]<<<-]>[-]>[<<+>>-]<<<<-]>>[<<+>>-]+<[-]<<<<-]>>>. 

Input is in the form of characters with code points of the numbers desired as inputs. Output is the same.

An explanation is coming when I have the time below. Do I get bonus points for not using exponentiation, multiplication, OR even addition?

Cell 3 (0-indexed) is the running total x. +<<+<<,<, Initialize tape with [n, a, 0, 1, 0, 1] [ While n: >[>+>>+<<<-]>[<+>-] Copy a 3 cells to right: [n, a, 0, x, a, 1] >[ While x: >[>>+>+<<<-]>>>[<<<+>>>-] Copy a 2 cells to right: [n, a, 0, x, a, 1, a, 0] <<[>[>+>+<<-]>>[<<+>>-]<<<-] Cell 7 = prod(cell 5, cell 6) >[-]>[<<+>>-]<<<<-] Move this value to cell 5. End while. >>[<<+>>-]+<[-]<<<<-] Update x to result of exponentiation. End while. >>>. Print the result! 

This works (tested) for x 0, 0 x, x 1, 1 x, x 2, 2 3, and 2 4. I tried 3 3 and 2 4, but each ran for several hours without finishing (in my Java implementation—probably not optimal). In theory, this works up to the cell size of the specific implementation.

#Br**nfuck, 128-5=123 bytes

+<<+<<,<,[>[>+>>+<<<-]>[<+>-]>[>[>>+>+<<<-]>>>[<<<+>>>-]<<[>[>+>+<<-]>>[<<+>>-]<<<-]>[-]>[<<+>>-]<<<<-]>>[<<+>>-]+<[-]<<<<-]>>>. 

Input is in the form of characters with code points of the numbers desired as inputs. Output is the same.

An explanation is coming when I have the time below. Do I get bonus points for not using exponentiation, multiplication, OR even addition?

Cell 3 (0-indexed) is the running total x. +<<+<<,<, Initialize tape with [n, a, 0, 1, 0, 1] [ While n: >[>+>>+<<<-]>[<+>-] Copy a 3 cells to right: [n, a, 0, x, a, 1] >[ While x: >[>>+>+<<<-]>>>[<<<+>>>-] Copy a 2 cells to right: [n, a, 0, x, a, 1, a, 0] <<[>[>+>+<<-]>>[<<+>>-]<<<-] Cell 7 = prod(cell 5, cell 6) >[-]>[<<+>>-]<<<<-] Move this value to cell 5. End while. >>[<<+>>-]+<[-]<<<<-] Update x to result of exponentiation. End while. >>>. Print the result! 

This works (tested) for x 0, 0 x, x 1, 1 x, x 2, 2 3, and 2 4. I tried 3 3, but it ran for several hours without finishing (in my Java implementation—probably not optimal). In theory, this works up to the cell size of the specific implementation.