Baby Sister Numbers

This is great! It reminds me a bit of ordinal arithmetic, in which addition is non-commutative. The ordinal numbers begin with all infinitely many natural numbers, followed by the first infinite ordinal, $\omega$. The next ordinal is $\omega +1$, which is greater than $\omega$. But $1+\omega$ is just $\omega$. 

Subtraction isn't canonically defined for the ordinals, so $\omega -1$ isn't a thing, but there's an extension of the ordinal numbers called the surreal numbers where it does exist. Sadly addition is defined differently on the surreals, and here it is commutative. $\omega \times -1=-\omega$ does exist though, and as with Norahats $\omega \times -1\times -1\times -1$ does equal $\omega$.

The surreals also contain the infinitesimal number $ϵ$, which is greater than zero but less than any real number. it's defined as the number between $0$ on the left and all members of the infinite sequence $1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\dots$ on the right. Not exactly Norklet ($\omega -1\ne ϵ$), but not too far away: ${\omega }^{-1}=\frac{1}{\omega }=ϵ$ :)

(h/t Alex_Altair, whose recent venture into this area caused me to have any information whatsoever about it in my head)

This is delightful. You should absolutely bring this up when she's older.

Is Noranoo a prime?

Is Norklet a prime?

Is the product of all primes below Noranoo, plus one, a prime?

What is the sum of all positive integers below Norklet?

Does Noranoo have a square root? Is the square root an integer?

Is Norklet a regular integer?  I'd think it has to be, unless there's a distinct counting system for distance from Noranoo (and Norahats).  Which just means that this is regular numbers, but with an overflow policy of staying at the endpoint.  

Can you get to Noranoo by +1 from 0?