TITLE: The largest number system
QUESTION [11 upvotes]: 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:

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?

REPLY [12 votes]: 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?