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 algebro-geometrically, as spectral schemes. In particular, examples of such spectra would be

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

2. Complex $K$-theory $\mathrm{KU}$;

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

4. 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:

1. The $p$-local Brown-Peterson spectrum $\mathrm{BP}$ and its connective covers $\rm{BP}\langle n\rangle$;

2. The Morava $K$-theories $K(n)$;

3. 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 algebro-geometric 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_2-rings.