Are norms intrinsically $\mathbb{R}$-valued? Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$?  
I have searched a fair bit (including in fairly encyclopedic textbooks), but have come up empty handed as to why everyone just uses $\mathbb{R}$ and does not consider generalizations.
The underlying motivation comes from looking at various theories of mathematics from a minimalistic, "universal algebra" perspective.  From that way of looking at things, as opposed to a more semantic perspective which focuses on applications of norms, it seems difficult to justify why norms must range over $\mathbb{R}$.  But perhaps it really is important that the range be Dedekind complete -- which would justify this choice.  But this is currently not apparent to me.
 A: The concept of valuation rings of arbitrary ranks exists. As a special case you get non-Archimedean valuations of higher ranks, whose corresponding norms are non-real.
Maybe you will find what you want from the following pages:
http://en.wikipedia.org/wiki/Valuation_ring (look for the notion of rank),
https://math.stackexchange.com/questions/1307/valuation-rings-of-rank-two has an example.
A: Not a norm based answer, but perhaps you may still find the work on "cone metric spaces" relevant---this dates back to 1934 by D. Kurepa (a student of M. Fréchet), who considered "metrics" that may take on a value in an ordered vector space. The paper linked to seems to present an updated view.
A: You might be interested in the whole (mostly Russian) literature on "Banach-Kantorovich" or "lattice-normed" spaces, which are:

"a triplet $(\mathcal U,E,\lambda)$ consisting of a vector space $\mathcal U$, a Dedekind complete vector lattice $E$ and a map $\lambda:\mathcal U\to E_+$ satisfying some natural conditions that allow one to consider $\lambda$ as a vector-valued analogue of the classical notion of a norm (...) Vector spaces equipped with a norm taking values in an Archimedean vector lattice were introduced by L. V. Kantorovich in 1935."

A: There is a very abstract generalization of norm and the general idea is as follows:
Valued field: consider a field $K$ and a valuation $|\cdot|:K\mapsto G\cup\{0\}$ satisfying


*

*$|x|\geq0$,

*$|x|=0$ iff $x=0$,

*$|x+y|\leq max\{|x|,|y|\}$,

*$|xy|=|x||y|$.


where $G$ is an arbitrary multiplicative ordered group and $0$ is an element such that $0<g$ for all $g\in G$. 
$G$-module: Let $G$ be a linearly ordered group. A linearly ordered set $X$ is called a $G$-module if there exists a map $G\times X\to X$, written $(g,x)\mapsto gx$, such that for all $g,g_1,g_2\in G$ and all $x,x_1,x_2\in X$ we have:


*

*$g_1(g_2x)=(g_1g_2)x$

*$1x=x$

*$g_1\geq g_2\Rightarrow g_1x\geq g_2x$

*$x_1\geq x_2\Rightarrow gx_1\geq gx_2$

*$Gx$ is coinitial in $X$

*$X$ has no smallest element.


Norm: Let $E$ be a vector space over $(K,|\cdot|)$ and let $X$ be a $G$-module. An $X$-norm on $E$ is a map $||\cdot||:E\to X\cup\{0\}$ such that for all $x,y\in E$, $\lambda\in K$:


*

*$||x||=0\Leftrightarrow x=0$

*$||\lambda x||=|\lambda|||x||$

*$||x+y||\leq\max\{||x||,||y||\}.$ 


For an introduction in this area I recommend the paper:
Banach spaces over fields with a infinite rank valuation - [H.Ochsenius A., W.H.Schikhof] - 1999
After that see: Norm Hilbert spaces over Krull valued fields - [H. Ochsenius, W.H. Schikhof] - Indagationes Mathematicae, Elsevier - 2006
