Apart from differentiating log (n²) from (log n)² in "log en squared", that at least makes sense without overloading the superscript-2's meaning. You cannot "square" a function (rather than the result of a function) and whilst I acknowledge that trig functions are so notated (at least with the likes of sin²(x), but not sin

^{-1}(x), and sin

^{-2}(x) could therefore be considered confusing), "iterate twice", i.e. "log of log of n", is really the only way I would understand this, without guidance to the contrary.

It doesn't help if it's

pronounced as "log squared en", as (unless we're breaking into perhaps some sort of Polish notation - thus log (n²)?), it really should be said as "log twice en" or nothing... Like log

^{-1} is "unlog n", if that weren't "(unstated base) to the power of n", anyways (also highlighting trouble with "log

_{b}^{2}(n)" notation), and the reciprocal of the log is often better written otherwise ("1/log n"? "

^{1}/

_{log n}"? "(log n)

^{-1}"? and also a proper TEX layout - according to availability in the medium being written in...).

I'll admit that I've never actually done much log of logging, or else squaring of logging, in a pure maths-theory environment where this may be necessary to understand the inconvenient precedents for (as per trigs). In coding, nested parentheticals make it clear(er!) what is meant, in a different manner. The "in the order of" notation of programming

theory just has never required me to delve into this inconvenient version of alternate applications of notation until now... I just have a sense that somebody, somewhere has got it wrong, and obviously it aten't me...