Note that $ p(x) $ has degree at most 2. Also, $ p(a) = p(b) = p(c) = 1 $. Thus, the polynomials $ p(x) $ and 1 agree at three different values, so by the Identity Theorem, they are the same polynomial. Hence, the degree of $ p(x) $ (which is the constant polynomial 1) is $ \boxed{0} $. You can check manually that \[p(x) = \frac{(x - b)(x - c)}{(a - b)(a - c)} + \frac{(x - a)(x - c)}{(b - a)(b - c)} + \frac{(x - a)(x - b)}{(c - a)(c - b)}\]simplifies to 1.