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, so it is a natural statement to consider.) A counterexample is given by $x^d-x-1$, whose discriminant has absolute value $d^d + (-1)^{d}(d-1)^{d-1}$.
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 $\mathbb{Q}(\zeta_n)$. Further examples of this kind would include 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 such subfields exist.
However, one may wish to restrict to reciprocal polynomials; for Lehmer's problem, this would be sufficient. Then I do not know a counterexample to that statement. (What about the polynomials $x^d - x^{d-1} - x +1$? I do not know how to calculate their discriminant. )