Where there ever number-systems popular that were not "base-of-10"?

Well we had 1999: “mille neuf cents quatre vingt dix neuf”
and ancient times were worse:

The name of fifteen-twenty is an ancient form of saying three hundred (15x20) when we counted in base twenty. 80 is another (4x20).

this might actually explain some of the weirdness in the electric systems of french cars …

Could something in quantum mechanics be highlighted if we used base 137?

For some reason, cardinal Welsh counting is pretty normal, at least in its modern form, but traditionally was vegidecimal. Lots of mixes of twenties and “ten-on-twenty” and so forth. Ordinal numbers still use the old word for twenty, hugain rather than dauddeg (the latter is literally two-twenties), then places a weird emphasis on fifteen. Eleven is unfed ar ddeg (first on ten), that goes up to fourteen pedwerydd ar ddeg (fourth on ten) and then a straight fifteenth with pymddegfed. Then you add to that: unfed ar bymtheg (sixteen, or first on fifteen) up through pedwerydd ar bymtheg (nineteen, or fourth on fifteen). Except, of course, for deunawfed (eighteen) which is literally two-ninth. Because reasons.

Two tens.

Sorry, yes, typo on my part, too late to edit.

On a family vacation decades ago, I picked up Teach Yourself Welsh. I didn’t get very far, but I loved the shear insanity of pedwar hugain ar deunaw for 99. Learning French much later, I saw that they were trying to keep up, but just couldn’t pull it off.

I’ve read speculation that the French partially base-20 system may be due to substratum influence from Celtic languages.

Welsh also has 16 as ‘one after fifteen’ and 17 as ‘two after fifteen’, as alternatives to ‘ten six’ and ‘ten seven’. If I spoke Welsh I would definitely use the ‘after fifteen’ variants.

Dialect differences are crazy in Welsh. There are, I think, five main dialect groups, but every town and village does something different from their neighbors twenty minutes away. Some towns never moved on from the traditional style of counting, for sure.

Is that why they’re base 20?

For 98, obviously

My previous understanding was that base 12 and 60 were also the results of finger counting:

A dozen by counting the phalanges of the four fingers with the thumb, and 60 by keeping track of five dozens with the other hand.

Not sure though how that tracks with the one left derivation but?

Anyone actually know?

You mean, at the far end of the sign listing the town’s name?

There are a number of unsubstantiated claims on the internet that the Ancient Sumerians, who used base 60, also finger counted in the way you specified. Some cites include the BBC’s History of Maths and Mathematics Magazine. The best one is an excerpt from Empire of the Sum: The Rise and Reign of the Pocket Calculator by Keith Houston (2023), published in Lapham’s Quarterly:

The argument is in the last 3 paragraphs of the article. Frankly I don’t follow it, but perhaps the book references the actual original research.

Abe Lincoln bought into this concept on at least one occasion…

I don’t know, but in case other me aren’t aware of this number popping up in strange places:

Were there really tribes that counted “one - two - three - many”?

The Pirahã, a tribe living in the Amazon, count “one - two - many”.

I do not think you are necessarily dealing with “tribes”. Only three numerals is hardly unheard of in various languages: Araona, Achagua, Baré, Hixkaryana, Martuthunira, Mangarrayi, Pitjantjatjara, !Xóõ

Supposedly this is not really correct; there are no numerals in Pirahã.

The words claimed to translate as “one” or “two” do not mean that; see
doi:10.1016/j.cognition.2008.04.007

Also no singular vs plural in that language.