Unused By Only Three

Question

First, beg for the most positive among the stables.

Now you can do the second, beg for the independent.

Now you can do the third, beg for the overlappable.

Now you can do the fourth, beg for the informations.

The first made you do the fifth, beg for the individual.

The first, second, and third made you do the sixth, beg for the green.

Meanwhile, do the seventh, beg for more than 602 sextillions.

What are the "three" in the title?

Subtle Hint:

65 = 5
75 = 4
109 = 2
115 = 1
13199 = 3
13253 = 6
13270 = 7

Moderate Hint:

67 = 15
70 = 111155/223
72 = 223/1155
74 = 223/11
78 = 23/11
83 = 11155/223
84 = 3/115
86 = 223/1115
87 = 223/111
8486 = 223/11155
13200 = /1
13217 = 22
13221 = 222
13223 = 2/1
13224 = 2/11
13225 = 3/112
13229 =
13230 = /1
13231 = /11
13251 = /1
13254 = 15/3
13257 = 22/11
13264 = 6
13267 = 6/22
13275 =
13276 = 22/11
13277 = 223/115
13278 = 23/1115
13279 = 5/2

Decisive Hint:

"The independent" is the fastest.

Decisive Hint 2:

The first one outside of "the stables" is from Poland.

Decisive Hint 3:

"The individual" is not among "the overlappable." But "the independent" is.

Decisive Hint 4:

Three countries, the answers are.

Critical Hint:

Conversion mistake was how one of these countries lost a rocket!

Answer 1

Partial Answer (mostly complete except for the subtle&moderate hints): following the hints by Jens and Stiv, the three are (it also explains Decisive Hint 4):

The United States, Liberia, and Myanmar, the only ones that do not use the metric system/SI units.

Critical Hint:

It refers to the famous Mars Climate Orbiter. "An investigation attributed the failure to a measurement mismatch between two software systems: metric units by NASA and non-metric (imperial or 'English') units by spacecraft builder Lockheed Martin."

To be a little blatant, I actually figured out the connection to SI units and chemical elements (to come later) before I read the hints :) But thanks for the hints still.

I will explain the 7 things in order, and talk about the hints when they become relevant in the discussion.

The seven things referred to in the text are

SI constants. Figuring out the SI system was such a red herring, since SI has exactly 7 base units. I spent a lot of time trying those, until I realized that there was 7 constants involved in the redefinition of SI base units in 2019.

and their symbols are simultaneously (or partially related to)

Symbols of chemical elements in the periodic table, with one "exception" for the fifth which represents an electron.

The first one is

Chemical Element: Cs or caesium, the single stable (i.e. no stable isotopes/"not radioactive") element with the least electronegativity, hence the most positive.

SI constant: $\Delta\nu_{Cs}=9192631770\ \text{Hz}$, the ground-state hyperfine structure transition frequency of the caesium-133 atom, used to define second. I included the number so that people might work on the subtle&moderate hints.

Regarding Decisive Hint 2:

I believe it refers to Po/Polonium, discovered by Polish scientists Marie&Pierre Curie. However, why is it the first? in terms of atomic number, Technetium/Tc is also unstable and precedes it; even if we disregard this artificial element and only consider the natural ones, Bismuth/Bi (the one before Po) "was discovered to be extremely weakly radioactive" in 2003. I have seen older textbooks that still use Po as the first one, though I personally think they should have been updated.

The second is

SI constant: $c=3\cdot 10^8\text{ m/s}$, the speed of light. In special relativity, it is independent of the chosen (inertial) reference frame.

Chemical Element: C/Carbon, a well-known non-metal element.

Decisive Hint:

The speed of light is currently known as the fastest physically possible speed.

The third:

Chemical Element: H/Hydrogen, a non-metal element. Non-metal elements can form covalent bonds, which requires orbital overlapping. (not 100% sure)

SI constant: $h=6.63\cdot10^{-34}\text{ J}\cdot\text{s}$, the Planck constant.

The Fourth:

SI Constant: $k=1.38\cdot10^{-23}\text{ J}/\text{K}$, the Boltzmann constant. It relates to entropy, and a kind of entropy is called information entropy in information theory. (not 100% sure)

Chemical element: K/Potassium.

The fifth:

SI constant: $e=1.6\cdot10^{-19}\text{ C}$, the elementary charge.

Chemical element: Electron/e/$e^-$.

"The first made you do the fifth" potentially refers to the fact that Cs is the element with the least electronegativity, so it is very easy to lose an electron. (not 100% sure)

Decisive Hint 3:

As explained above, hydrogen and carbon are nonmetals, while the electron is not even an element.

The sixth:

SI constant: $K_{cd}=683\text{ lm}/\text{W}$, the luminous efficacy of monochromatic radiation of the green light, used to define the unit cd (candela).

Chemical element: Cd/Cadmium.

"The first, second, and third made you do the sixth": this is the only part where I am unable to figure out what's going on.

The seventh:

SI constant: $N_A=6.02\cdot10^{23}\text{ mol}^{-1}$, the Avogadro constant. The numerical value is indicated directly in the text.

Chemical Element: Na/Sodium.

EDIT: in response to the hint that

chemical elements

are mostly irrelevant, I'll add few preliminary thoughts (before I work on it more seriously).

Overlappable may refer to bosons, as multiple particles can be in the same quantum state (roughly speaking), in contrast to fermions. Photons are bosons, and their energy is given by $E=h\nu$ where $h$ is the Planck constant, explaining the third. Presumably this also explains why the independent is overlappable since it also refers to photons. Electrons are fermions and more than one of them cannot be in the same quantum state (Pauli exclusion principle), which explains why the individual electrons mentioned in the fifth are not overlappable.

Answer 2

Everyone else did most of the brilliant deduction and have some insights into the hints that got us here (e.g. conversion mistake) but here is what may be closer to a final answer?

You are the

SI base units of the International System of Units (SI), or more broadly, the SI itself/metric system.

There are seven SI base units (with associated base constants) which appear to match the seven clues. These are the base of the metric/SI system used in all but three countries (Credit Stiv). Details are here: https://en.wikipedia.org/wiki/SI_base_unit, what follows is a summary of key points matching them to clues.

First, beg for the most positive among the stables

This is 's', the second – The definition of a second is derived from Cs (Caesium), the stable (non-radioactive) element with the least electronegativity (most positive) – credit Tesla

Now you can do the second, beg for the independent.

This is 'm', the metre. It is defined by taking the speed of light in a vacuum where nothing else exists (independent). Hint 2 “this is fastest” Maybe because it involves the speed of light which is the fastest thing. Whereas say the definition of ‘second’ came from a transition frequency.

Now you can do the third, beg for the overlappable.

This is 'kg', the kilogram. Perhaps because it is expressed using a lot of other unit types (J s equal to kg m2 s-1) so is considered 'overlappable'. Hint 3 says ‘m’ is among the overlappable, and sure enough, m is part of the definition here of a kg, and m itself uses the second as part of its own definition. In any case, this is kg by process of elimination as it’s the only one left when solving the others.

Now you can do the fourth, beg for the informations.

This is 'K', kelvin. A common synonym for information is Knowledge, keying K

The first made you do the fifth, beg for the individual.

This is 'A', the ampere. Individual keys “one” and A is ordinal one in the alphabet. Another possible interpretation is that its associated base constant is related to the elementary charge e, which is noteworthy in that it is indivisible (individual). And the 'first made you beg for the fifth' again refers to the fact that the elementary charge is defined in terms of amperes and seconds (second was the first unit in this list).

The first, second, and third made you do the sixth, beg for the green.

This is cd, the candela. As noted by Tesla, the definition of a candela is related to the radiation of green light. Similar to kg, this is expressed in terms of units "kg m s" which are the first, second, and third. So another overlappable.

Meanwhile, do the seventh, beg for more than 602 sextillions.

This is 'mol', the mole. As noted by Tesla and a clear break in to this puzzle, 602 sextillions relates to the Avogadro constant and the definition of mole.

Not positive if all these are using the interpretation intended by the clues, but if we interpret the Subtle Hint as UNICODE = ORDER, it straight up reads the above, validating all results. E.g. Unicode 65 = A and so forth.

So what are the 3 countries which do not use these (title question?) Stiv has given this, but here it is in conclusion:

United States, Liberia, and Myanmar (Burma)