Solution:

Answer: 4

If all the polynomials had real roots, their discriminants would all be nonnegative: $a^{2} \geq 4 b c$, $b^{2} \geq 4 c a$, and $c^{2} \geq 4 a b$. Multiplying these inequalities gives $(a b c)^{2} \geq 64(a b c)^{2}$, a contradiction. Hence one of the quadratics has no real roots.

The maximum of 4 real roots is attainable: for example, the values $(a, b, c) = (1, 5, 6)$ give $-2, -3$ as roots to $x^{2} + 5 x + 6$ and $-1, -\frac{1}{5}$ as roots to $5 x^{2} + 6 x + 1$.