Obstructions for $E_n$-algebras In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. 

Have the obstructions for an object of an $\infty$-operad $\mathcal{C}^\otimes$ from having an $\mathbb{E}_n$-algebra structure been studied? More precisely, what conditions must an object $\mathscr{C}$ of $\mathcal{C}^\otimes$ satisfy to be an object of $\mathrm{Alg}_{/\mathbb{E}_n}(\mathcal{C}^\otimes)$?

 A: I think this question is too general. Already the $1$-categorical version is hard: when does an object in a monoidal category have a monoid structure? When does it have a commutative monoid structure? Even in $(\text{Ab}, \otimes)$ I don't know a reasonable general statement that can be made here (that is, I already don't know good necessary and sufficient conditions for an abelian group to have a compatible ring structure, or a commutative ring structure).
Here is a nontrivial theorem in the setting of compact abelian groups: any compact topological ring is totally disconnected. 
Is there a particular case you're interested in? 
A: Let me expand a little on what Qiaochu and Craig mentioned.
If you want an obstruction theory for building an uber-gadget, you'll need (i) an algebraic approximation to such gadgets, and (ii) a way to resolve every uber-gadget by special uber-gadgets where the algebraic approximation remembers everything about the thing you started with.
Condition (i) usually takes the form of a functor $Uber \rightarrow Approx$ with some nice properties, and then for (ii) you hope that maybe there is an adjoint to this functor, and that $Uber$ is generated under sifted colimits by the essential image of this adjoint (I'm sweeping everything important and techncal under the rug). 
Once you have this, you can start trying to formulate the problem of starting with an object in $Approx$ and adding structure until you get to an object in $Uber$ (this shouldn't be too surprising: you've essentially required that $Uber$ is monadic over $Approx$ so that elements of $Uber$ are like elements of $Approx$ with extra structure). If you do this very carefully, you'll have expressed the space of objects lifting a given one. All of this is developed for $E_\infty$-ring spectra in Goerss-Hopkins and you can find write-ups here: http://www.math.northwestern.edu/~pgoerss/ . I'm pretty sure the same proofs give you an $E_n$-obstruction theory, and this appears elsewhere as well. 
Now, you probably already knew all that. But I said it to remind you that there's just no hope of proceeding if you don't have (ii) and for (ii) you need your algebraic approximation to be strong enough. So you shouldn't ask: "Given a spectrum, what's the space of $E_\infty$ ring structures on is?" You need to have a little bit to get going, some candidate for the Dyer-Lashof operations or (in the Goerss-Hopkins case) a commutative comodule algebra over $E_*E$ for a suitable cohomology theory, $E$. Otherwise your answer will probably be ridiculous and uncomputable (this is sort of what Qiaochu was pointing out.) 
