Jacob Lurie has extensively developed derived algebraic geometry in the setting of $ \mathbb{E}_\infty$ ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in particular) a way to view some of the spectra topologists care about algebrogeometrically, as spectral schemes. In particular, examples of such spectra would be

The complex cobordism spectrum $ \mathrm{MU}$ ;

Complex $ K$ theory $ \mathrm{KU}$ ;

The spectrum $ \mathrm{TMF}$ of topological modular forms;

The sphere spectrum $ \mathbb{S}$ .
On the other hand, there are many important spectra which don’t admit $ \mathbb{E}_\infty$ structures (and hence don’t fit in Lurie’s SAG) such as:

The $ p$ local BrownPeterson spectrum $ \mathrm{BP}$ and its connective covers $ \rm{BP}\langle n\rangle$ ;

The Morava $ K$ theories $ K(n)$ ;

The Ravenel spectra $ X(n)$ .
There has been work on SAG over $ \mathbb{E}_n$ rings, in particular by John Francis, in his thesis. Focusing on $ n=2$ , what results of SAG are expected to be troublesome to extend to the setting of $ \mathbb{E}_2$ ring spectra?
I’m also tempted to ask here Sanath’s question on this topic:
[…] what are some results in either the purely algebrogeometric or purely chromatic aspects of spectral algebraic geometry which rely upon using the entire $ \mathbb{E}_\infty$ ring structure?
to be found in his A love letter to E_2rings.