It doesn't make much sense to analyze “je n'ai jamais vu personne” as a triple negative. It's either a single negative ne qualified by jamais and personne, or two negatives jamais and personne with the reinforcement ne. Negation in French is currently evolving from the first model to the second model. In both models, the word ne has a different role from jamais and personne.
Historically, negation in French followed the same principles as other Romance languages, with ne being the main negative particle. Negation could sometimes be qualified by a noun or adverb, e.g. “ne … jamais” = “not … ever”, “ne … personne” = “not … anyone” (literally: “not … [a] person”), etc. French acquired a peculiarity that it gradually became mandatory to qualify all negations, even for yes/no statements. If there was no “natural” qualifier, point would do, and then around the late 19th century to early 20th pas became the default qualifier. This is not a double negation, since the qualifier itself does not convey negation, but a single negation in two parts.
This model allows for more than one qualifying adverb. For example, “je n'ai jamais vu personne” = “I have not ever seen anyone” has a single negation ne with both a temporal qualification and an animate quantification.
Today, French is going through an evolution where ne can be dropped and the sentence still has a negative meaning. From the perspective of early modern French, this is in some sense the opposite of a double negation — you could call it a nullary negation, since negation is expressed without a negative word. In fact, the adverbs that once had a positive meaning have acquired a negative meaning of their own. For example, rien was originally a noun meaning “thing”, but it now almost always means “nothing”. Personne still means a person when used as a noun, but means “nobody” when used on as a pronoun or adverb. In modern colloquial French, you can say “j'ai vu personne”, meaning “I haven't seen anyone” (this is common spoken French but not considered correct in formal contexts when written).
If you analyze formal French negation by the lights of colloquial French, you could say that “je n'ai vu personne” is a double negation with negative meaning. It's possible to combine multiple negative adverbs other than pas, as in “je n'ai jamais vu personne”, and the sentence remains negative. You could thus say that “je n'ai jamais vu personne” is a triple negative. In colloquial French, “j'ai jamais vu personne” is a double negation with negative meaning.
However, pas works differently. If you put pas in a sentence, it negates other negative adverbs. This is not standard French: in standard French, you can't combine pas with another negative adverb. But if you say something like “?je n'ai pas vu personne” or “?j'ai pas vu personne”, it's like “I haven't not seen anyone”: a double negative in the sense that the negation is negated, i.e. I have seen someone and I'm saying so in a whimsical way.
The fact that ne, pas, and the other qualifying negatives work in three different ways makes it problematic to count one ne and two qualifying negatives as three. They just don't add up in this way. Standard formal French has a single negation ne which can't be repeated. Standard colloquial French has multiple negative words which can be sorted in two categories: pas which can't be repeated or combined with another negative, and others (rien, personne, jamais, etc.) which can be combined together but not with pas, forming double (or more, e.g. “j[e n]'ai plus jamais vu personne”) negations with a negative meaning.
See also https://french.stackexchange.com/questions/20096/should-the-2nd-negative-particle-be-defined-positively-only/20097#20097 for more a bit more details about negative adverbs in French and their evolution.