We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors
\begin{align*}
    (-)^\times   &\colon \mathsf{Ab} \to \mathsf{Sets},\\
    \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets}
\end{align*}
given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

[A similar approach](https://mathoverflow.net/a/404304) in the $\infty$-world gives the $\mathbb{E}_\infty$-groups $QS^0$ and $\Omega Q\mathbb{RP}^\infty$. Passing to spectra via the [equivalence between $\mathbb{E}_\infty$-groups and connective spectra](https://mathoverflow.net/questions/280980), we obtain the sphere spectrum $\mathbb{S}$ and a spectrum $E$ corresponding to $\Omega Q\mathbb{RP}^\infty$ (one possible name for $E$ would be "$\mathbb{S}/2$" since it has an analogous universal property to $\mathbb{Z}/2$, but that notation already usually denotes the mod 2 Moore spectrum). For comparison, their first $8$ homotopy groups are as follows:
$$
\begin{aligned}
\pi_0(\mathbb{S}) &\cong \mathbb{Z},\\
\pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\
\pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\
\pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\
\pi_4(\mathbb{S}) &\cong 0,\\
\pi_5(\mathbb{S}) &\cong 0,\\
\pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\
\pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5,
\end{aligned}
\quad\quad
\begin{aligned}
\pi_0(E) &\cong \mathbb{Z}/2,\\
\pi_1(E) &\cong \mathbb{Z}/2,\\
\pi_2(E) &\cong \mathbb{Z}/8,\\
\pi_3(E) &\cong \mathbb{Z}/2,\\
\pi_4(E) &\cong 0,\\
\pi_5(E) &\cong \mathbb{Z}/2,\\
\pi_6(E) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\
\pi_7(E) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2.
\end{aligned}
$$
(The ones for $E$ are taken from [Liulevicius](https://www.jstor.org/stable/1993859?seq=1#metadata_info_tab_contents); see also [MO 230790](https://mathoverflow.net/questions/230790).)

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What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $E$?