on first reading, I missed that second 'not' -- if you meant to put it, that's not correct; if you didn't mean to put it, that's still not correct.
I looked to the third 'not' and was stymied because I hadn't seen the second either.
Nevertheless, notwithstanding, still, however, yet, are some alternate adverbs that could make the statement nonetheless emphatic.
In the the sentence "I cannot accept your offer, but I thank you none the less," it simply means I thank you despite the fact that I don't want what you're offering. I thank you anyway. I thank you even though I don't need to thank you. It's just an idiom. Idioms don't need to analysed in detail to try to make sense of them. They don't make sense. That's what makes them idiomatic.
Equivocal language. When olly uses nonetheless in "adverbs that could make the statement nonetheless emphatic," he should have used "none the less" since that's not what 'nonetheless' means. He meant not less than using 'nonetheless.' But 'nonetheless' as one word does not mean not less than. Its meaning transcends its etymology. It means 'anyway.'
