This weekend's solution to the 31-digit puzzle is indeed,
if you feed 2^100 to the program 'bc':
1267650600228229401496703205376

It's 2 to the 100th power, denoted as 2¹⁰⁰ (it doesn't look as nice as I would like), mostly written in English as:
1,267,650,600,228,229,401,496,703,205,376
and it's a 1 followed by another 30 digits, in three groups of ten digits:
1 2676506002 2822940149 6703205376.
In the US number naming system, the number is:
one nonillion, 267 octillion, 650 septillion, 600 sextillion, 228 quintillion, 229 quadrillion, 401 trillion, 496 billion, 703 million, 205 thousand, 376.
In the Dutch naming system (source) it is:
1 quintiljoen, 267 quadriljard, 650 quadriljoen, 600 triljard, 228 triljoen, 229 biljard, 401 biljoen, 496 miljard, 703 miljoen, 205 duizend 376.
When and where did this go wrong? (*1)

And, as we know, 10¹⁰⁰ = googol, a one followed by a hundred 0s.

SOLUTION
(I've used a smaller font for some of the more detailed answers, perhaps you could use CTRL+'+' (enlarge) and CTRL+'-' (decrease) the text in your browser)

┌───┐
│ 1 │ 1. First decimal of pi.
├───┤ The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. 
│ 2 │ 2. First prime number.
├───┤ A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. 
│ 6 │ 3. VI.
├───┤ (in Roman numerals)
│ 7 │ 4. First decimal of e (or Euler's number).
├───┤ The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828. 
│ 6 │ 5. First perfect number.
├───┤ In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. 
│ 5 │ 6. Square root of 25.
├───┤
│ 0 │ 7. Not positive, not negative.
├───┤
│ 6 │ 8. Number of faces of a cube.
├───┤
│ 0 │ 9. Last digit of the number of squares on an International draughts game board.
├───┤ (100 squares)
│ 0 │ 10. Number of elements in an empty set.
├───┤
│ 2 │ 11. Pair.
├───┤
│ 2 │ 12. Even prime number.
├───┤
│ 8 │ 13. Square root of the number of squares on a chess game board.
├───┤ (64 squares)
│ 2 │ 14. Atomic number of helium.
├───┤
│ 2 │ 15. Number of planets between Earth and Sun.
├───┤ (Venus, the second planet from the Sun, and Mercury, the nearest planet to the Sun)
│ 9 │ 16. Square root of the number of squares in the Letter Sudoku.
├───┤ ((3+3+3)×(3+3+3) = 81 squares)
│ 4 │ 17. The name of my team on World Community Grid: top…0.nl.
├───┤
│ 0 │ 18. Name of pointer in C indicating that it doesn't point to any valid object.
├───┤ (null pointer)
│ 1 │ 19. Number of Earth's natural satellites.
├───┤ (The Moon, with a diameter about one-quarter that of Earth (comparable to the width of Australia))
│ 4 │ 20. Smallest semiprime (or biprime).
├───┤ In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. 
│ 9 │ 21. 1³ + 2³.
├───┤ (1 + 8)
│ 6 │ 22. Number of legs of an insect.
├───┤
│ 7 │ 23. Sum of two opposite faces on a six-sided dice.
├───┤
│ 0 │ 24. Exit code of the command true, as opposed to false, on UNIX/Linux.
├───┤
│ 3 │ 25. First Mersenne prime.
├───┤ In mathematics, a Mersenne prime is a prime number that is one less than a power of two. 
│ 2 │ 26. Base of the binary system.
├───┤ The base-2 or binary numeral system is a positional notation with a radix of 2. It uses only two symbols: typically "0" (zero) and "1" (one). 
│ 0 │ 27. You cannot divide by ....
├───┤
│ 5 │ 28. Number of faces on a square pyramid.
├───┤
│ 3 │ 29. Number of South Africa's capitals.
├───┤ (Pretoria (executive), Cape Town (legislative), Bloemfontein (judicial))
│ 7 │ 30. Last digit of James Bond's code name.
├───┤ (007)
│ 6 │ 31. ... Lake (name of this lake at 46°42'37"N 95°46'43"W, near Five Lake).
└───┘ (Six Lake)

When I was looking for a description by which a six would have to be put into the last cell, I was thinking about 6 letters: "Sgt Joe", but how…? No
I decided it should have something to do with the state Minnesota - and what do you know? - on Wikipedia I stumbled across "Six Lake, a lake in Minnesota".

Thanks for your participation!
Adri

[*1] American billion, British billion (source)
The origin of this misery lies in France. The word million was used there in the fifteenth century. The mathematician Nicolas Chuquet needed words for bigger numbers and came up with a clever system. Billion was a million squared (based on the Latin bis, meaning two), a tryllion was a million to the third, and a quadrillion was a million to the fourth. Handy, because that way you could go on endlessly.
A few decades later, the mathematician and linguist Jacques Peletier devised another system on top of this. Henceforth, the suffix -ard stood for "thousand times", i.e., a thousand million is a milliard, a thousand billion is a billiard, and so on.
This system was enthusiastically adopted throughout Europe. All of Europe? No, one country persisted in using different words, and that was France itself. To the French, a million was 10⁶, a billion 10⁹, a trillion 10¹², and so on.
Very inconvenient, but it took until 1961 for the country to adapt to the rest of Europe. Meanwhile, the former French system had long since spread across the United States. After a while, the British were so tired of the confusion between the American billion and British billion that they joined their transatlantic ally in 1974. And so the European unity in numerals soon came to an end.

PS "super power(s)" was a brilliant hint. (superscript)
Did anyone notice "100" percent?

Adri,

I had noticed the "100%" at the time, and just managed to avoid replying to it with "More power 2 you..." :-)

Once again, thanks for the puzzle (and confirmation of the answers)

Cheers - Al.

This one was erg makkelijk compared to the last one. The only one I had to look up was the Dutch motor sports event. This is quite similar to the New York Times minicrossword, which luckily they allow you to do without having a subscription.
Also a very interesting name for the ice skates. I take it they are different from schaatsen,
hockey schaatsen, kunstschaatsen.
Cheers

There is a homophone for the first answer. A remembrance of my youth.
You should be able to see it here


Cheers

Sgt.Joe:Ah, I've been there in my youth with my father. I remember motor racer Wil Hartog, well-known by his nickname 'The White Giant' thanks to wearing a white racing outfit, where black was normal, in combination with his length; he worked at his father's grass-drying establishment. My father bought dried grass there to resell to the farmers, for their cattle and herds.

Let's have a look.
From Wikipedia: [one skate] consists of a shaped length of wood secured to a metal blade of the same length to form a single unit which can be bound to a boot or shoe. On the Dutch Wikipedia page for Friese doorlopers an image is shown of a pair of old wooden skates that aren't Friese doorlopers and on the English Wikipedia page the same image is being displayed where they say these are Friese doorlopers. Typical for a Friese doorloper is that 'It was based on earlier designs of wooden skate, and differs from them in that the blade extends several inches behind the heel instead of ending under it. This reduces the risk of the wearer falling over backwards, particularly when stopping. (With the older designs, the wearer braked by raising their toes and digging their heels into the ice; with the new one, by angling the blades.)'

Adri
EDIT: I have translated the designated Dutch 'Friese doorloper' Wikipedia image caption and put that in the English one on Wikipedia.
EDIT: Here is a photo of some Friese doorlopers I found on Pinterest:

and here are some on Flickr: