I think it helps here to be concrete, not abstract.
Think about the support of such transformations. ln $x$ requires that $x > 0$, ln ln $x$ requires that $x > 1$, ln ln ln $x$ that $x > e = \exp(1)$, and so forth. Can you think of variables that naturally satisfy such limits? Also, if one limit is natural, the others won't be.
So, while you have a mathematical family here, it is not statistically very useful.
Also, even if by accident or otherwise your data satisfy these limits, these are very severe transformations, even ln repeated twice. They may be the ultimate outlier tamers for large positive values, but they also may have the opposite effect of creating outliers by separating arbitrarily small values just above the lower limits.
If you want something stronger than the logarithm for transforming highly skewed positive values, the (negative) reciprocal is the best candidate. It has the added virtues that its units of measurement are easy to think about and its interpretation is often easy (e.g. times and rates are reciprocals of each other). (A negative sign is optional here if preserving order is important.)
A quite different and much more positive point is that functions such as the loglog $-$ln($-$ln($x$)) and complementary loglog ln($−$ln$ (1−x))$ can be very useful for $0 < x < 1$ and appear as link functions for generalized linear models.