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?
It seems hard enough to construct non-trivial non-trivial integer polynomials having high degree $d$ and discriminant bounded by $d^d$ (the cyclotomic polynomials have this property, but we may consider them to be trivial). There could be a finite set $T \subset \mathbb{Z}[x]$ such that, up to a translation and a reflection, all integer polynomials having non-zero discriminant of absolute value $\leq d^d$ would factor into products cyclotomic ones and polynomials from $T$.
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 $\leq (cd)^d$ are essentially cyclotomic, does not hold without additional assumptions. Since we may take $c = M(f)$ as Mahler proved, this is a natural statement to consider: its truth would have strengthened Lehmer's conjecture.
A counterexample is given by $x^d-x-1$, whose discriminant has absolute value $d^d + (-1)^{d}(d-1)^{d-1}$. We may take more generally any sequence of irreducible trinomials with $\pm 1$ coefficients and degree going to infinity. This follows by the explicit calculation of the discriminant of the general trinomial; the statement and simple derivation of the formula is given as Theorem 2 in Swan's 1962 paper Factorization of polynomials over finite fields in the Pacific Journal of Mathematics. (Another reference is Prasolov's book Polynomials, which reproduces the same calculation).
An example of a rather different kind (in particular, having unbounded Mahler measure) is provided by 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 would include the minimum polynomials of the generators over $\mathbb{Z}$ of the integer rings of other bounded index monogenic subfields, if such exist, of either $\mathbb{Q}(\zeta_n)$ or the splitting fields of the $\pm 1$ trinomials considered in the previous paragraph.
However, one may wish to restrict to reciprocal polynomials; for Lehmer's problem this would be sufficient. For these the statement seems very interesting, and could well be true; by the formula in Swan's article, it certainly holds for all trinomials. The above examples are of no use here: the only reciprocal $\pm 1$ trinomials are $x^{2n} \pm x^n +1$, and those have only cyclotomic factors.