A more consistent definition of tetration of tetration for rational exponents

[UVIR]

... then we have a more serious problem.
If you define \( {^0x} \) to be different from 1, say \( {^0x}=a \), then \( {^1x}=x^a \). And in common use of the word tetration \( {^1x}=x \) (hopefully you dont want to change this for the sake of continuity of your construction).
So if you accept this then any (real) value of \( a \) different from 1 poses the contradiction \( x^a=x \). You see that it is necessary to define \( {^0x}=1 \).

I see your point. My definition forces:

\( {^0}x=x^{1/x}\\
{^1}x=x^{x^{1/x}}\\
{^2}x=x^{x^{x^{1/x}}}
\)
...
which, I guess goes against standard notation. I am leaving for vacations tomorrow, so I will try to examine the function defined as above and see if there's anything more interesting about it. If there is, I will report it back in a week or so.

Thanks to all who participated.