This question showed the following:
言わない (negative)
言わなく・ない (negative+negative)
言わなく・なかった (negative+negative+past)
I've never seen this conjugation before. Why would you have a negative + negative? Where is that coming from? How does that work exactly?
In addition to the previous answer, often these forms are seen with an particle in the middle (は or も), and are used followed by for such as が・けど (examples borrowed/stolen from internet, any translation mistakes my own)
気持ち分からなくはないけど... It's not that I don't understand his feelings, but... (I do understand, but I still don't approve of his actions/won't change my mind/etc)
まあ、言わなくもないけど… 時と場合によるよ... Well, it's not that I wouldn't say that... it depends on the situation... (I would say that... but maybe not at this time/in this case)
Double negatives are used not just in Japanese.
It's not that I'm not hungry...
Since (-1)•(-1)=1, it makes only sense to use a double negative, if its meaning is different from the positive, viz. either stronger or weaker than the positive.
In English, the double negative feels weaker than the positive. In Japanese, the double negative is stronger than the positive. Maybe because negation is in general felt to be quite strong. (To me) it makes sense, then, that double negation would be stronger than no negation.
I don't think double negation is used often to give a strong positive statement. But the negative question, like
お茶を飲みに行かない？
is likewise applied to negative forms, e.g.
あまりおいしくなくない？
Don't you (also) think it's not very good?
Of course the idea of negating a negative can be extended ad infinitum, like in the phrase これよくない？よくないこれ？よくなくなくなくなくせいいぇい (YouTube).
Caveat: Sometimes the double negative can have the same meaning as the negative. E.g.
よくない？
asks whether something is good. Then the answer
よくなくない。
can mean "not good".
I guess the better strategy is to rely on other factors than the number of negations...
