Question: What are some ring spectra $E$ satisfying the following conditions?
The coefficients $E_\ast = \pi_\ast(E)$ are graded-commutative;
There is a Kunneth spectral sequence $E_\ast(X) \otimes_{E_\ast} E_\ast(Y) \Rightarrow E_\ast(X \wedge Y)$ for finite spectra $X,Y$;
The coefficients $E_\ast$ form a graded-artinian ring (i.e. satisfies the descending chain condition for homogeneous ideals).
Clearly there is a canonical sequence of examples $E = H\mathbb Q, K(1), K(2), \dots, H \mathbb F_p$. More generally, if $E_\ast$ is a graded field then $E$ is an example -- but this essentially reduces to the examples listed.
I'm particularly curious if the telescope $T(n)$ is (can be chosen to) satisfy (1,2,3).