Why must polynomials have degrees that are positive integers? What's the benefit of knowing 'this' is a polynomial? And why should we care?
 A: Polynomials are everywhere infinitely differentiable (not to mention defined and continuous!), and positive-degree polynomials in one variable are completely factorisable into linear factors over $\Bbb C$. If we allow negative powers, the example $\frac{1}{x}$ is undefined at $0$ and has no roots in $\Bbb C$. Thus polynomials as we currently define them are a very nice set of functions to consider.
A: Polynomials are obtained by means of addition and multiplication alone. That gives them many interesting properties such as continuity, differentiability, null $d+1^{th}$ derivative, ring algebra, known number of roots, connection to linear algebra via ODE's and Eigenvalues and many many more.

Generalized polynomials with negative integer powers can be rewritten as the ratio of a polynomial over a power and are of little interest ($x+4x^{-3}=\dfrac{x^4+4}{x^3}$). Rational fractions, i.e. the ratio of two polynomials, are richer and more important.

Generalized polynomials with rational exponents can also be turned to ordinary polynomials by a change of variable (e.g. $x^{3/2}+4x^{1/3}=y^9+4y^2$ with $x=y^6$). So nothing really new.

Finally, generalized polynomials with real exponents are not attractive. They cannot be defined on negative arguments (and fit poorly with complex numbers) and don't enjoy the composition property "$p(q(x))$ is a polynomial". E.g.
$$(x+1)^\pi$$ cannot be written as a generalized polynomial in $x$.
