Is there a general bound on a polynomial's root with the largest norm?
When Rouche's theorem is used, it still seems that the polynomial's root with the largest norm still needs to be found if we want a tight bound.
The Eneström-Kakeya theorem is helpful if the coefficients of the polynomial in question are all real and positive. Essentially, the zeros lie within the circle whose radius is the largest ratio of consecutive coefficients of the polynomial. Here is a good reference (PDF).
There are some generalizations, for instance when the coefficients lie in a given sector symmetric about the positive real axis. I could provide further references if this is what you're looking for.