TITLE: How universal is operadic approach to studying algebras?
QUESTION [10 upvotes]: I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain operad. For example, associative algebras are algebras over the associative operad $\mathcal{As}$, commutative algebras - over the commutative operad $\mathcal{Com}$. The same is true for Lie, Poisson, Gerstenhaber, $A_{\infty}$-algebras.
So my question is: is there a (interesting) class of algebras that can't be viewed as a class of algebras over a certain operad? I mean, how universal is the operadic approach to studying algebras?
If there are algebras that are not algebras over an operad, then what are the obstructions?
Thank you very much!

REPLY [2 votes]: I put in a remark about this question back in 2013. I'm going to expand on that for emphasis.  Let me shamelessly quote myself, from the expository paper http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf
where I describe the coining of the word operad.
``To these ends, I consciously sacrificed all-embracing generality: many types of algebras defined by identities are deliberately excluded".  In particular, algebras defined by identities with repeated variables are deliberately excluded despite their evident interest. I knew about alternating algebras and Jordan algebras from a course at Yale with Nathan Jacobson and about graded Lie algebras (char 2 or 3)  from my thesis, and they are deliberately excluded.