Polynomials and L^p(R) As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)?  To be more precise, there are no non-trivial polynomials in that space and, to me, polynomials are not only the simplest functions, they are the building blocks of most everything which can be (easily) manipulated algorithmically.  And restricting to a compact support is really a non-answer, since one of the great things about polynomials is that they are global, analytic functions.
To ask a more precise question: are there some spaces of (total, real-valued) functions which are both nice from a functional analysis point of view, and contain all the polynomials?
 A: Distributions (or tempered distributions).
A: You are referring to $L^p(\mathbb{R}, \mathcal{B}, \mu)$ in the case that $\mathbb{R}$ is endowed with Lebesgue measure $\mu$. Consider instead the measure $\nu$ given by $d\nu = f d\mu$, where $f$ is in the Schwartz space and $f$ does not take the value zero. Because the product of a polynomial with $f$ is also in the Schwartz space, and the Schwartz space is contained in $L^p(\mathbb{R}, \mathcal{B}, \mu)$, it follows that polynomials are in $L^p(\mathbb{R}, \mathcal{B}, \nu)$.
A: If one wants to do algebra, or symbolic computation, then polynomials are indeed the simplest type of function.  But if one wants to do analysis, or numerical computation, then actually the best functions are the bump functions - they are infinitely smooth, but also completely localised.   (Gaussians are perhaps the best compromise, being extremely well behaved in both algebraic and analytic senses.)
That said, I'm not sure what your question is really after.  If you want a function space that contains the polynomials, you could just take ${\bf R}[x]$.  Of course, this space does not come equipped with a special norm, but polynomials, being algebraic objects rather than analytic ones, are not naturally equipped with any canonical notion of size.  Due to their growth at infinity, any such notion of size would have to be mostly localised, as is the case with the weighted spaces and distribution spaces given in other answers.
