The largest number system If my number set construction memory doesn't fail me (I'll edit if errors are pointed out), we start out with Peano's axioms to get to $\mathbb{N}$, and in the need of an additive inverse for its elements, construct $\mathbb{Z}$. Then, in order to be able to invert nonzero integers with respect to multipilication, $\mathbb{Q}$ is created. For there to be inexact integer roots of rationals, the field $\mathbb{R}$ is constructed, and so that every real number has integers roots, $\mathbb{C}$ is devised. These questions arise:


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*What kind of operation — and number — becomes possible by constructing quaternions and octonions?

*The hierarchy of the cardinalities of these sets is $\#\mathbb{N} = \#\mathbb{Z} = \#\mathbb{Q} < \#\mathbb{R} = \#\mathbb{C}$. How are $\#\mathbb{H}$ and $\#\mathbb{O}$ inserted in it?

*Can yet another number set be constructed from $\mathbb{O}$?

*Does the said hierarchy stop at some number set — that is, is there a largest number set?
 A: I will give a point which was amiss in both the answers and somewhat connects this question to the set theoretic tags it has.
There can be a largest number system, in the sense of ordered fields (that is it embeds $\mathbb R$ but not $\mathbb C$) and that is The Surreal Numbers.
It is a class field, which means it is not a set and has no cardinality. As an order it embeds all the ordinals and every ordered field.
I would have to admit that I am far from having a real clue about this number system, so instead I will link to two MO questions that might be somewhat helpful:


*

*Where do surreal numbers come from and what do they mean?

*What's wrong with the surreals?
A: Vhailor's answer takes care of most of your questions. I'll try to help out with the rest.
I'm not sure what it means for a mathematical concept to have a purpose, but I would say the purpose of $\mathbb{H}$ and $\mathbb{O}$ is that, together with $\mathbb{C}$ and $\mathbb{R}$ itself, they are the only finite dimensional normed division algebras over $\mathbb{R}$. It's true, the quaternions have a fair number of applications to modeling rotation and whatnot, but (IMO) we are interested in normed division algebras over $\mathbb{R}$, the fact that we have this classification (Hurwitz's theorem) of the finite dimensional ones is sufficient reason to single them out and study them. Hamilton was trying to construct a 3-dimensional normed division algebra over $\mathbb{R}$, until he realized it couldn't be done and realized (on the now-famous bridge) that he had to move up to 4.
We have $\dim_\mathbb{R}(\mathbb{R})=1$, $\dim_\mathbb{R}(\mathbb{C})=2$, $\dim_\mathbb{R}(\mathbb{H})=4$, and $\dim_\mathbb{R}(\mathbb{O})=8$. Technically there are also the sedenions $\mathbb{S}$, which are 16-dimensional over $\mathbb{R}$. They do not form a finite-dimensional normed division algebra over $\mathbb{R}$, which is why they don't appear in the classification given by Hurwitz's theorem. 
There is no "largest number set". Mainly because what it means for something to be "a number" is not a rigorous (or particularly useful) mathematical notion.
A: I think I can answer questions 2, 3, and 4.
For 2, since $\mathbb{H}$ and $\mathbb{O}$ respectively have the same cardinalities as $\mathbb{R}^4$ and $\mathbb{R}^8$, they have the same cardinality as $\mathbb{R}$. (taking cross products of infinite sets with themselves doesn't change their cardinality).
For 3 and 4, Hurwitz's theorem tells us that the only normed division algebras over the reals, up to isomorphism, are the four ones that you mentionned.
Edit : Not as much an answer as the others, but for 1, I know that one of the motivations for quaternions is that, since complex numbers make studying rotations in the plane so easy, we want to construct an algebraic framework to study rotations in higher dimensions (3 and 4).
