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).