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Please don't question my curiosity.

Mod edit: please read the title carefully; any move that the Pokemon learns through level up is excluded from this question, as are all Pokemon that learn 0 TMs.

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For example, Ivysaur can learn 26 TMs. One of them, solar beam, can also be learned by level up, so Ivysaur's "TM-but-not-level up count" is 25. I want to know which species has the least nonzero "TM-but-not-level up" count.
By the way, Bulbasaur can't learn any TM moves by level up, so wherever you got that information is wrong.
Well, Kakuna Beldum Ditto and Magikarp learn 0.
But the lowest number that can still learn TMs is 2, Hidden Power and Protect on Burmy.
So I did my spreadsheet thing and apart from the 0 TM Pokemon, the most incompetent Pokemon are Wynaut, Wobbuffet and Spewpa, who learn 1 TM, as well as Tynamo and Burmy, who learn 2 TMs each. Unfortunately all those Pokemon learn their TM moves through level-up, which makes this task significantly more difficult since apart from these Pokemon, the least competent Pokemon is Pyukumuku, who learns 17 TMs. It's not reasonable to assume that Pyukumuku and other equally-incompetent Pokemon would have the lowest TM-but-not-level-up move count since their TM count is already relatively high, so that makes answering this question that extra bit more difficult. Yay.
So are you saying this spreadsheet can't order Pokemon from least to most TMs?
It just can't order Pokemon from least to most TM-but-not-level-up moves. Currently.
I'm gonna have heaps of spare time now so I might be able to find a way to make it happen. It wasn't as easy as I thought it would be.

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I'm not gonna bother explaining how I did this (my Excel document is glitching out and there are black holes on the page lol) but basically the five Pokemon who learnt the lowest amount of TMs (Wobbuffet, Wynaut, Spewpa, Burmy, Tynamo) all learn their TMs through level-up, so this question became significantly more difficult to answer. But we have reached a conclusion! The Pokemon who can learn the least TM-but-not-level-up moves in Sun & Moon are...

14 moves - Pyukumuku
16 moves - Kartana
17 moves - Trapinch & Snorunt
18 moves - Rockruff

Compare this data to Pokemon who learn the least amount of TMs:

1 move - Wobbuffet, Wynaut & Spewpa
2 moves - Burmy & Tynamo
17 moves - Pyukumuku
19 moves - Eevee
20 moves - Shelmet & Kartana

When only looking at how many TM moves each Pokemon learned, Pyukumuku sat above Burmy and Tynamo as the Pokemon who learnt the fourth-lowest amount of TMs (including Pokemon who didn't learn TMs). Essentially, Pyukumuku retains this spot, thus rendering my efforts useless... but there were some notable changes. All of Eevee's TM moves are exclusively TM moves, thus Eevee was saved from reaching the bottom of the number-of-TM-but-not-level-up moves ranking! Clearly, the asker's modification in criteria changed the rankings up quite a bit.

Okay, I hope I'm done now.

Hope I helped. :)
Source (literally the best thing ever) and Excel

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If anyone cares:
13% of Pokemon learn none of their TM moves through level-up.
21% of Pokemon learn 1 of their TMs through level-up.
22% of Pokemon learn 2 through level-up.
17% of Pokemon learn 3 through level-up.
13% learn 4 through level-up.
7% learn 5 through level-up.
19 Pokemon learn 6 TM moves through level-up.
8 Pokemon learn 7 TM moves through level-up.
7 Pokemon learn 8 TM moves through level-up.
Delphox learns 9 TM moves through level-up.
Fizz, this person didn't read the question properly. Can you hide the answer.