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."