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 pretty 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.

Even if we restrict to reciprocal polynomials (which for Lehmer's problem is fine), I would bet that polynomials such as $x^d - x^{d-1} - x +1$ would be counterexamples. But I do not know the formula for that polynomial's discriminant.