To reiterate Serre's question (Open Problem 2.5 in Odlyzko's survey): Must there be only finitely many polynomials having root discriminant below a given bound?
With this answer I just want to note that the much stronger statement formulated in the last paragraph of user631's answer, which concerned the existence of a $c > 1$ such that all irreducible polynomials of discriminant $< (cd)^d$ are essentially cyclotomic, cannot be true. (By Mahler's inequality, this would have strengthened Lehmer's conjecture). A counterexample is given by $x^d-x-1$, whose discriminant is $d^d + (-1)^{d}(d-1)^{d-1}$. Another example: the minimum polynomial of the generator $\zeta_n+\zeta_n^{-1} = 2\cos(2\pi/n)$ of the integer ring of the maximal totally real subfield of $\mathbb{Q}(\zeta_n)$. Additional examples would be the minimum polynomials of the generators over $\mathbb{Z}$ of the integer rings of other bounded index monogenic subfields of $\mathbb{Q}(\zeta_n)$, if those exist.