Let $P\_n(t)$ be polynomials with integer coefficients with $d\_n = \deg(P\_n(t))$ going to infinity when $n$ goes to infinity
and with nonzero discriminants $disc(P\_n(t)) \neq 0$.


Question: Is
$$
\lbrace disc({P\_n(t)})\rbrace ^{\frac{1}{d\_n}}
$$
bounded when $n$ goes to infinity ?



Old question of Serre on discriminants of a sequence of polynomials
Some people seem a little confused by the wording of this question.
The better way of phrasing the question is: Is there a *lower* bound on
the root discriminants of *polynomials* of degree $d$ as $d \rightarrow \infty$.
The survey paper by Odlyzko (noted by Gerry in the comments above):


<http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1990__2_1/JTNB_1990__2_1_119_0/JTNB_1990__2_1_119_0.pdf>


mentions this as a question of Serre and others (it is a pretty obvious question to ask), and also points out that essentially nothing is known about this question. I don't think the situation has changed considerably in the intervening $20$ years.


As Franz notes, the existence of class field towers is no way implies the existence of polynomials with bounded discriminant - "most" fields are not monogenic.


Some have wondered - perhaps optimistically - whether asymptotically there is even a bound
$$|\mathrm{disc}(P)| > (cd)^d$$
For irreducible $P$, where one can take $c$ to be some fixed constant $ > 1$ unless $P$ is essentially a cylotomic polynomial. If this was true, then it would imply that the Mahler measure of an algebraic integer that is not a root of unity is bounded away from $1$, answering a question of Lehmer. 


