In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree $d$. My question is what is provably known about the asymptotic behavior of this function $f(d)$, and in particular, whether or not it is known to be exponentially bounded in the degree $d$.
Regarding heuristics (I would be interested in good ones, too), let me only mention that the global function fields model, when appropriately formulated, does have an exponentially growing $f(d)$, but that this is probably a poor guide in this type of question. (For instance, since the function field model admits a fully explicit and monogenic construction, which is conjectured to not exist in the case of number fields: the root discriminant of an integer irreducible polynomial is widely believed to approach infinity as the degree grows.)
As everyone knows, the Golod-Shafarevich towers give plenty of examples of number fields with an exponentially bounded discriminant, but those by their construction have solvable Galois groups, and are irrelevant in my question.
Added example. It is easily seen that $f(d) < d^d$, but this crude bound, I suppose, would be far from the truth. To see this bound, recall that Selmer proved that the trinomial $t^d - t - 1$ is irreducible with maximal Galois group $S_d$, and of discriminant $\pm( d^d - (1-d)^{d-1})$, which gives the bound, at least for $d$ odd (and the slightly weaker bound $f(d) < 2d^d$ in general).
