An important example in homotopy theory is given by operadic twisting. This was introduced by T. Willwacher in <a href="https://doi.org/10.1007/s00222-014-0528-x">"M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra"</a> (Inventiones Math. 200 (2015), 671–760), and its comonadicity was proved by V. Dolgushev and T. Willwacher in <a href="https://doi.org/10.1016/j.jpaa.2014.06.010">Operadic twisting – With an application to Deligne's conjecture</a> (J. Pure Appl. Alg. 219 (2015), Issue 5, 1349-1428). 

Essentially, for a dg operad $\mathcal{O}$ equipped with a morphism from the operad $\mathrm{Lie}$ (or $\mathrm{Lie}_\infty$), one can construct a new operad $\mathrm{Tw}(\mathcal{O})$. It is done in two steps. First, one considers the operad $\mathrm{MC}(\mathcal{O})$ whose algebras are $\mathcal{O}$-algebras with a given solution to the Maurer--Cartan equation $$d\alpha+\frac12[\alpha,\alpha]=0.$$ 
Then one "twists" the differential $d_{\mathrm{MC}}$ of that latter operad by adding the "commutator with the commutator" (the operadic commutator with the unary operation $\ell_1^\alpha=[\alpha,-]$). It turns out that this construction is a functor and, moreover, a comonad, and furthermore, coalgebras over this comonad are precisely operads $\mathcal{O}$ for which algebra structures can be twisted by solutions to the Maurer--Cartan equation. This construction has been used in a variety of contexts (from algebraic topology and deformation theory to, more recently, stochastic PDEs).