The discussion about double negatives made me feel that the term itself
is not especially correct.

It seems to me that actually two strategies in the treatment of negation
are opposed:

1) Next negation 'magnifies' the previous one; speaking in such language,
you can pile up as many negations as you wish, without restriction to just
'double'. E. g., in Russian

Ja nikogda nichego     nikomu     ne  govoril
I  never   nothing+Gen nobody+Dat NEG

- is a perfectly correct sentence, meaning, roughly, 'I've never told
anything to anybody'.

2) Next negation negates the previous one; in theory, an even number of
negations will be equal to affirmative, an odd number - to simple
negative, with some additional shade of meaning. But typically even two
negations in one phrase are exceptional, and more then two, eventually
forbidden. E. g. in Latin:

Nunquam nil     narravit
Never   nothing told+3.Sg.

'It has never happened that s/he'd tell nothing' = 'S/he has always told
(at least) something'.

I wonder if _Nunquam nil non narravit_ or _Nunquam nemini nil narravit_
are possible in Latin, and what such utterances could mean...

The situation in English is in a way transitional. Most speakers are aware
of <dia/idio>lects that allow 'heaped negations' which are considered
incorrect. Therefore, even two negations in one phrase are avoided.

_Je n'ai pas dit rien  personne_ - is this possible in any variety of
native French? If not, the situation in French is better termed 'split
negation', IMHO.