From Tom Leinster:

Since I asked the question I've found a few examples; they've all got the same flavour about them, so I'll just do my favourite.

If V is the commutative monoid, then a V-enriched category is a set A plus two functions [-,-,-]: A x A x A ---> V [-]: A ---> V satisfying [a,c,d] + [a,b,c] = [a,b,d] + [b,c,d] [a,a,b] + [a] = 0 = [a,b,b] + [b] for all a, b, c, d.

A few remarks: In the case where V is an abelian group, the first axiom already implies the other two if we define [a] = -[a,a,a]. Namely, by letting a=b in the first axiom, it follows that [a,a,c] is independent of c. If V is an abelian group, then one can get an example of the above structure from an arbitrary map {-,-} : A x A ---> V by letting [a,b,c] = {a,b}+{b,c}-{a,c} and [a] = -{a,a}. Tom's "area" example is of this form. In fact, if V is an abelian group, then *any* example of a V-enriched category is (non-uniquely) of the form described in the previous paragraph: Fix some x in A (if any), and define {a,b} = [a,b,x]. What about the non-group case? In general, [a,b,c] need not always be invertible in V. In fact, [a,b,a] need not be invertible. For a simple example of this, let V be the natural numbers and define [a] = 0, [a,b,c] = 0, if a=b or b=c, 1, if a,b,c pairwise distinct, 2, otherwise (i.e., if a=c but a,b distinct). This indeed works. Best wishes, -- Peter Selinger

The example: let A be a subset of the plane. Choose a smooth path P(a,b) from a to b for each (a,b) in A x A, and define [a,b,c] to be the signed area bounded by the loop P(a,b) then P(b,c) then (P(a,c) run backwards); also define [a] to be -(area bounded by P(a,a)). (There's meant to be an orientation on the plane, so that areas can be negative.) Then the equations say obvious things about area - don't think I'm up to that kind of ASCII art, though.