Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi\_r^S(S^0)$?



How nilpotent is the ring of stable homotopy groups of spheres?
Let $\alpha \in \pi\_s(S^0)$ for $s > 0$ be an element of positive degree in the stable stems. Then $\alpha$ has positive-dimensional filtration in the Adams-Novikov spectral sequence: in other words, it is annihilated by complex bordism. 


The $E\_\infty$-page of the Adams-Novikov spectral sequence for the sphere is known to have a vanishing "curve" of slope tending to zero as $t -s \to \infty $ (this is essentially the Devinatz-Hopkins-Smith nilpotence theorem). The exact behavior of this curve is not known, but there is a conjecture of Hopkins (from [this talk](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.hha/1251832464&page=record)) that it looks like $s \simeq \sqrt{t-s}$. In this case, any element in $\pi\_t$ would have nilpotence order $O(t)$, by considering the spectral sequence. This conjecture is (as far as I know) unproven. 


It might be worth noting that explicit nilpotence-type results have been proven using power operations (this is the sort of technique that led to Nishida's proof that order $p$ elements are nilpotent), for instance the Toda relation in [this paper](http://projecteuclid.org/euclid.pja/1195521243).


