The wedge sum $\bigvee_{k \in 2 \mathbb{Z}} S^{k}$ is an $A_\infty$ring spectrum: the connective cover is the free $A_\infty$ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the element in $\pi_2$. It is also homotopy commutative. Can it be made into an $E_\infty$ring spectrum? More generally, given an $E_\infty$ring spectrum $R$, when can $\bigvee_{k \in 2 \mathbb{Z}} \Sigma^k E$ be made into an $E_\infty$ring?

Ah, tracked it down. Here is an argument. I should mention that Peter once pointed me towards an original source due to McClure, or page 238, Prop. 6.1, of SLN 1176 (the $H_\infty$ book). Suppose you had such a ring object $R$. We examine its mod2 homology. This has several features:
In particular, these together would say $$ 0 = P_2 Q^4 t = \binom{2}{2} Q^{2}P_0 t + \binom{2}{0} Q^{4}P_2 t = Q^2 t = t^2. $$ If $E$ is a commutative $MU$algebra, then you can use the map from $MU$ to its periodic version $MUP$ to produce $R \wedge_{MU}MUP$, which is 2periodified. Unfortunately, we know very few genuine $MU$algebras. Barry Walker proved that complex Ktheory is one of them, based on Matthew Ando's work on studying $H_\infty$ structures on LubinTate cohomology theories. My understanding is that the problem is still open for almost all of the LubinTate cohomology theories. 

