$\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in the sphere spectrum $\mathbb{S}$.

As discussed in Section 2 of Ulrich–Nikolaus's Twisted differential cohomology, such an object may be defined as a symmetric lax monoidal functor $\mathbb{S}\to\mathsf{Sp}$, where $\mathbb{S}$ is considered as a monoidal $\infty$-groupoid.

In particular, given an $\mathbb{S}$-graded ring $R$, it follows that the ring $\pi_0(R)$ is $\mathsf{Ho}(\mathbb{S})$-graded. Since moreover $\mathsf{Ho}(\mathbb{S})\cong\tau_{\leq1}(\mathbb{S})$, such a $\tau_{\leq1}\mathbb{S}$-graded ring consists of the data of a $\mathbb{Z}$-graded ring $R_{\bullet}$ whose multiplication map $$\mu_{n,m}\colon R_n\otimes R_m \to R_{n+m}$$ is commutative if $nm$ is even, and otherwise satisfies $\sigma_{n+m}(ab)=ba$.

So, are there any interesting/non-trivial "in nature" examples of $\mathbb{S}$-graded ring spectra?