Things that should be positive integers...really? Kronecker. Nuff said. Even the numbers themselves historically started 
as positive integers and were subsequently generalized to hell and back.
Here are some other well known concepts that "should" involve $\mathbb{N}$ 
but were generalized to $\mathbb{Q}$, $\mathbb{R}$ or even $\mathbb{C}$:


*

*Dimension $\rightarrow$ Hausdorff dimension.

*Factorial $\rightarrow$ gamma function.

*Differentation $\rightarrow$ half-differentation (etc.)


So, can you extend this small to a big list? 
(Motivation: Some hypothetic knot polynomial I calculated with
demanded a dimension of its associated group representation 
- thus the "rt" tag - of 60/11. That is noooooot boding well 
for its existence. :-)
 A: It's interesting to note that this has happened with several notions of "dimension". Krull dimension of rings has been extended to notions as GK-dimension, for example.
As a complementary answer... what would be a ring of characteristic $-\pi$?
A: Let $f:\mathbb{Z}_p\to\mathbb{Z}_p$ be a "nice" map on the $p$-adic integers (or a map on some more general space with a $p$-adic topology). People who study $p$-adic dynamcis investigate what the iterates of $f$ do to points of the space. So if we fix a point $\alpha\in\mathbb{Z}_p$, we can define an iteration map
$$
  I  : \mathbb{N} \longrightarrow \mathbb{Z}_p,\qquad
    I(n) = f^n(\alpha).
$$
The map $I$ is naturally defined on $\mathbb{N}$, and if $f$ is invertible, then it clearly extends to $\mathbb{Z}$. But for various applications, one would like to evaluate $I(n)$ for $n\in\mathbb{Z}_p$. So the example is 


*

*iteration an integral number of times $\to$ iteration a $p$-adic number of times. 


A very pretty application of this idea is in the paper:
Bell, J. P. ;  Ghioca, D. ;  Tucker, T. J.  The dynamical Mordell-Lang problem for étale maps.
 Amer. J. Math. 132  (2010),  no. 6, 1655--1675.
A: (Probably should count towards your Dimension example) 
Sobolev Spaces of Integer Dimension $\rightarrow$ Sobolev(–Slobodeckij) Spaces of fractional Dimension
Has Important applications for numerically solving boundary integrals
A: The writhe is the fundamental differential geometric invariant of a closed space curve. I think it is the most useful topological invariant outside mathematics- biologists use it to study circular DNA molecules, and chemists use it in the study of long polymers. For space curve $C(t)$ it's defined as the double integral
$\frac{1}{4\pi}\int_{C\times C}\frac{C^\prime(s)\times C^\prime(t)\cdot (C(s)-C(t))}{|C(s)-C(t)|^3}ds dt.$
but most people think of it as the number of positive crossings minus the number of negative crossings. This quantity is naturally an integer. The integral formula is based on the Gauss integral for the linking number, but has a complicated history, with a lot of contribution from non-mathematicians.
But, what to do, most real-life long molecules aren't closed space curves. And so biologists, chemists, and physicists, followed by mathematicians, generalized the writhe to open space curves. The idea is that writhe makes sense for a tangle diagram, so they integrated over all projection angles of the open space curve. The result is a definition for the writhe of an open space curve, which is a real number (which can be efficiently estimated). I think it's differential geometry's most useful real numbers for studying open space curves where they occur in biology, chemistry, and physics.
A nice survey of writhe in various contexts is Berger and Prior's The writhe of open and closed space curves.
A: The natural extension of Euler characteristic to orbifolds is valued in Q.
A: A natural generalization of cardinality of sets is groupoid cardinality, which is a real number.
A: Motivic integration, where the underlying measures are valued in rings of motives.
