Working from right to left.

[bo198214]

Doing it from right to left would also make it easier as going from x^^y to x^^(y+1) would only involve putting x^^y to the power of x just like going from x^y to x^(y+1) only involves multiplying x^y by x and going from x×y to x×(y+1) only involves adding x to x×y.

Well if you do it from left to right, there does not come up something new. If we have
x^^0=x
x^^(y+1)=(x^^y)^x

then must x^^y = x^(x^y).