Does an ordered field $F$ satisfying the Cauchy-Cantor axiom but not the Archimedean axiom exist?
Cauchy-Cantor axiom: Every system of nested intervals $I_1 \supset I_2 \supset \cdots \supset I_n \supset \cdots$ have a common point. Here, an interval $I$ is any set of the form $I= \{ x \in F \mid a \leqslant x \leqslant b\}$ for some $a\le b$ in $F$.
Archimedean axiom: For every $x,y \in F$ with $y>0$ there exists a unique $n \in \mathbb{Z}$ such that $ny \leqslant x < (n+1)y$.
An ordered field that satisfies the Nested Interval Property need not be Archimedean.
A very nice paper that answers all questions of this general character that one might ask, and more, is James Propp's Real Analysis in Reverse.
For an early class of examples, please see the classic Rings of Continuous Functions (Gillman and Jerison).
There are also constructions based on ultrapowers of the reals.