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 refined 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: $2^{-d}T_d(2x)$, the minimum polynomial of the generator $\zeta_d+\zeta_d^{-1} = 2\cos(2\pi/d)$ of the integer ring of the maximal totally real subfield of $\mathbb{Q}(\zeta_d)$. Other examples would include all additional monogenic abelian fields (subfields of $\mathbb{Q}(\zeta_d)$), if those exist.