Are $\frac18$-th root transformations recommended? My colleague wants to analyze some data after transforming the response variable by raising it to the power of $\frac18$ (that is, $y^{0.125}$). 
I'm uncomfortable with this, but struggling to articulate why. I can't think of any mechanistic rationale for this transformation. Nor have I ever seen it before, and I worry that perhaps it inflates Type I error rates or something -- but I have nothing to support these concerns! 
Additionally, my colleague finds that these transformed models outperform untransformed models in an AIC comparison. Does this, in itself, justify its use?
 A: It is common practice to apply power transformations(Tukey, Box-Cox) with arbitrary values on the response. From that perspective, I see no particular concern regarding your value of 1/8 - if that transformation gives you good residuals, go for it.
Of course, any transformation changes the functional relationship that you fit, and it may be that 1/8 doesn't make sense mechanistically, but that would not be a concern to me when the purpose is not to extrapolate or fit parameters of a physical law, but to get a proper p-value on the sign of the effect (I'd argue that's the normal use case in a regression). For that purpose, your only concern is that the function fits to the data in the domain of your predictor values (wrt mean and residual variation), and that is easy to check. 
If you are unsure about the best value for the power transformation, and want to compare between different options, you should not directly compare AIC / likelihood values because the power transformation changes the scale of the response. Fortunately, it turns out that it's relatively straightforward to calculate a correction for the transformation, such that different transformations can be compared via their (corrected) likelihood (see, e.g. here).
In R, this is implemented in MASS::boxcox - this is a convenient way to choose the right value for the power. 
