The one value of super square root function A function can not have more than one value.
i.e. we only take the positive value for $y$ where $y=\sqrt 4$
But what about the super square root?, if both value are positive real values like:
$$y=\sqrt {\left({\frac12}\right)^{\frac12}}_s$$
In this case we have two real values $y=\frac12$ , $y=\frac14$
I wonder what value for $y$ should I accept in the following case:
$$y=\sqrt{a}_s  (e^{-\frac1e}<a<1)$$ 

I would use the function 
$$ \operatorname{ssqrt}(x) = \exp( W( \log( x  ))) $$
where $W$ is the Lambert-W (at its zero'th branch) and $x$ is in the range $ \exp(-\exp(-1))\approx 0.6922 \ldots 1$ as reference/as principal value.       
This gives for $x= 1/2^{1/2}\approx 0.707 $
$$ \operatorname{ssqrt}(x) = 0.5 =1/2 $$
... for $x= 1/3^{1/3}\approx 0.693 $
$$ \operatorname{ssqrt}(x) \approx 0.403542672016 \approx 1/2.478 $$
... for $x= 1/4^{1/4}\approx 0.707 $
$$ \operatorname{ssqrt}(x) = 0.5 =1/2 $$
and so on
