The Degree of Zero Polynomial.

One also wants $\deg(P+Q)\leq\max(\deg P,\deg Q)$ to hold, even if $P=-Q$.

Added much later: and maybe more importantly, in Euclidean division of some polynomial $A$ by $B\neq0$ we want the remainder $R$ to satisfy $\deg(R)<\deg(B)$, even if the division is exact (i.e., if $R=0$).

Convention 0. Marc has already explained why $\mathrm{deg}(0) = -\infty$ is a good convention.

Convention 1. On the other hand, if we want $\mathrm{deg}(PQ) \geq \mathrm{deg}(Q)$, well this implies $\mathrm{deg}(0) \geq \mathrm{deg}(Q)$ for all $Q$, so therefore $\mathrm{deg}(0) = +\infty$ is a good convention. Just take a look at the following divisibility sequence, which suggests that $\mathrm{deg}(0)$ should be "as large as possible."

$$1 \mid x \mid x^2 \mid x^3 \mid \ldots \mid 0$$

Bottom line: don't assume the reader understands your own personal preferred conventions regarding $\mathrm{deg}$ unless and until you've told them.

Convention 2. By the way, if you're thinking of $\mathbb{R}[x,y]$ as a graded algebra, you probably want the homogeneous polynomials of each degree to form an $\mathbb{R}$-module. In this case, its best to think of $0$ as having every possible degree, so that it belongs to each $\mathbb{R}$-module. So we might write

$$\mathrm{deg}(0) = \{0,1,2,3\ldots\},$$

or say something like "degree isn't a function, its a relation."

If you define the degree of a polynomial as the degree of the highest non-zero power, then if the polynomial is zero, the degree is undefined. You could define it by convention, to make sense of some general rules, and then as the other answers explain, you get different results.

Now, think of why $x^0 =1$. The proof is by using the laws of exponents, $\frac{x^n}{x^n} = 1 = x^{n-n}$. However, $0^0$ is undefined, since $x^y$ is not continuous at 0 (as a function of two variables). However, we often define $0^0=1$. It is a convention, to make some multiplications easier.

Similarly, the degree of the zero polynomial is not defined, but we use different conventions to be able to generalise formulas (like the product of polynomials).