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.
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