In most cameras  or lists, ISO increases by doubling the prior amount with intermediate steps of 1/3

Yes, ISO scales by doubling, which matches how light works in photography. Each time the ISO number doubles (for example, from ISO 100 to ISO 200), you are increasing the sensor's sensitivity by exactly one full "stop" of light, making the image twice as bright.
To give photographers finer control, modern digital cameras introduce 1/3-stop intermediate steps between those major doubles.

The Standard ISO Scale (With 1/3 Steps)

Here is how a standard ISO progression looks on most modern cameras. The full stops (doubles) are bolded, and the 1/3 intermediate steps sit between them:
• 100
	• 125 (+1/3 stop)
	• 160 (+2/3 stop)
• 200 (Twice as bright as 100)
	• 250 (+1/3 stop)
	• 320 (+2/3 stop)
• 400 (Twice as bright as 200)
	• 500 (+1/3 stop)
	• 640 (+2/3 stop)
• 800 (Twice as bright as 400)
	• 1000 (+1/3 stop)
	• 1250 (+2/3 stop)
• 1600 (Twice as bright as 800)

Why Use 1/3 Intermediate Steps.

If you only changed your ISO by full standard doubles, a single click could make your image too bright or too dark. The 1/3-stop steps allow you to make micro-adjustments to your exposure.
This precision lets you perfectly balance your image brightness without being forced to change your aperture (which alters your depth of field) or your shutter speed (which alters motion blur).

So, for 1/3 stop steps, every three steps result in a doubling of the ISO value

Exactly. Every three steps results in a doubling of the ISO value, which equals one full stop of light.
Because ISO scales logarithmically rather than linearly, the camera doesn't add the exact same number each time. Instead, it multiplies the previous value by approximately 1.26 (the cube root of 2, or \(2^{1/3}\)) for each 1/3-stop step.
Here is exactly how those three steps compound to double your value:
• Starting point: ISO 100
• Step 1 (+1/3 stop): \(100 \times 1.26 \approx\) ISO 125
• Step 2 (+2/3 stop): \(125 \times 1.26 \approx\) ISO 160
• Step 3 (+3/3 or 1 full stop): \(160 \times 1.26 \approx\) ISO 200 (The exact double)
This same "three clicks to double" rule applies no matter where you start on the dial. For example, three clicks up from 800 takes you past 1000 and 1250 straight to 1600.

Is there any rule in the photographic literature that says "ten clicks to multiply by ten?

There is no official rule in photographic literature explicitly stated as "ten clicks to multiply by ten," but the math of the standard 1/3-stop camera system makes this [almost] true.
Because photography math is logarithmic rather than linear, this phenomenon is a natural byproduct of how cameras are engineered.

The Math Behind the 10 Clicks

As established, a single click on a camera dial set to 1/3-stop increments scales the exposure value by multiplying it by \(2^{1/3}\) (approximately 1.2599).
If you click the dial 10 times, you compound that multiplier ten times:
\(\text{Multiplier}=(2^{1/3})^{10}=2^{10/3}=2^{3.3333}\)
When you calculate \(2^{3.3333}\), the result is 10.079.
Because camera manufacturers slightly round off fractional ISO values for simplicity (using 125 instead of 125.9, or 160 instead of 158.7), exactly 10 clicks on your dial will multiply your ISO or shutter speed by exactly 10.

Seeing It in Action on the ISO Scale

If you count exactly 10 clicks up from your base ISO, you will land directly on its 10x counterpart:
• Start at ISO 100 → Click 10 times → Land on ISO 1000
• Start at ISO 200 → Click 10 times → Land on ISO 2000
• Start at ISO 400 → Click 10 times → Land on ISO 4000

Seeing It in Action on Shutter Speeds

This trick works even better for calculating extreme neutral density (ND) filter exposures with shutter speeds. If you are trying to slow down your shutter speed:
• Start at 1/1000s → Click 10 times slower → Land on 1/100s
• Start at 1/250s → Click 10 times slower → Land on 1/25s
While you won't find this written as a formal "law" in standard photography textbooks, it is a highly useful mathematical shortcut used by manual shooters to quickly shift exposures in their heads.