(as a sidenote on terminology, as Jonathan points out in the comments, "spectrum" is not really a good name for your objects precisely because you expect them not to be spectra)
If you look at theorem 8.8 in the GGN paper you cited, you'll see that categories with a nicely behaved tensor product yield examples by taking their groupoid core.
Here are some explicit classical examples:
For a commutative ring spectrum, $\mathrm{Proj}_R^\simeq$ is a commutative semiring space, whose ring completion is $K(R)$ when $R$ is connective. More generally, if $R$ is $E_k$, this semiring space "is" $E_{k-1}$.
There is an equivariant version of $\mathbb F$, namely $\mathrm{Fin}_G^\simeq$ for a (pro)finite group $G$; as well as an equivariant version of the previous example, where for a commutative ring $G$-spectrum $R$, you call "projective" any $R$-module which is a summand of a direct sum of $R\otimes G/H_+$'s (no shifts allowed)
Here's an example which doesn't directly fit into the context of [Thm 8.8, GGN] (although it can be made to, via e.g. the condensed/pyknotic approach):
$\mathrm{Vect}_\mathbb R^\simeq$, $\mathrm{Vect}_\mathbb C^\simeq$, the groupoids of finite dimensional vector spaces over $\mathbb{R,C}$ respectively, but where you take into account the topology on the hom-sets. Note that this is not literally $(\mathrm{Vect}_\mathbb K)^\simeq$ for some $\infty$-category of vector spaces and all linear maps obtained by applying the nerve to the topological category - indeed, if you do that then all mapping spaces are contractible and so you get trivial categories; so you have to restrict to isomorphisms before passing to $\infty$-categories. Their ring completions are $\mathrm{ko, ku}$ respectively.
Again, the above example has an equivariant version.
(the equivariant versions can be made into semiring $G$-spaces, but let me not get into that here)
There are probably tons of other examples.
One of the default examples of ordinary graded commutative rings is the polynomial ring $\mathbf Z[t]$. Let us first examine the analogue of that, and then see where else that leads!
1. $S$-grading on $S\{t\}$
For the sake of clarity, allow me to denote the underlying infinite loop space (equivalently: grouplike $\mathbb E_\infty$-space) of the sphere spectrum $S$ by $\Omega^\infty(S)$.
Recall that, by the Barrat-Priddy-Quillen Theorem, the ininite loop space $\Omega^\infty(S)$ can be described as the group completion of the groupoid of finite sets and isomorphisms $\mathcal F\mathrm{in}^\simeq$. Consider the constant functor $\mathcal{F}\mathrm{in}^\simeq \to\mathrm{Sp}$ with value $S$ - since the latter is the monoidal unit, this is a symmetric monoidal functor. Left Kan extension along the canonical map $\mathcal F\mathrm{in}^\simeq \to (\mathcal F\mathrm{in}^\simeq)^\mathrm{gp}\simeq \Omega^\infty(S)$ produces a symmetric monoidal functor $\Omega^\infty(S)\to \mathrm{Sp}$, and as such exhibits an $S$-graded $\mathbb E_\infty$-ring.
But which one? To figure out which one, note that the passage to the "underlying $\mathbb E_\infty$-ring" of an $S$-graded $\mathbb E_\infty$-ring is given by passage to the colimit. Thus the underlying $\mathbb E_\infty$-ring, which we have just adorned with an $S$-grading, is
$$
\varinjlim_{\Omega^\infty(S)}\mathrm{LKan}^{\Omega^\infty(S)}_{\mathcal{F}\mathrm{in}^\simeq}(S)\simeq \varinjlim_{\mathcal F\mathrm{in}^\simeq}S,
$$
where we used that the left Kan extension of a functor does not change its colimit. Now we can use the explicit description of the groupoid of finite sets $\mathcal F\mathrm{in}^\simeq \simeq \coprod_{n\ge 0}\mathrm B\Sigma_n$ to obtain
$$
\varinjlim_{\mathcal F\mathrm{in}^\simeq} S\simeq \bigoplus_{n\ge 0}S_{h\Sigma_n}.
$$
We may recognize this as the free $\mathbb E_\infty$-ring on a single generator $S\{t\}$, corresponding in terms of spectral algebraic geometry to the smooth affine line $\mathbf A^1$ (as opposed to the flat affine line $\mathbf A^1_\flat = \mathrm{Spec}(S[t])$ for the polynomial $\mathbb E_\infty$-ring $S[t]\simeq \bigoplus_{n\ge 0}S$).
2. $S$-grading on symmetric algebras
The previous example can be easily generalized by observing that $\mathcal F\mathrm{in}^\simeq$ is the free $\mathbb E_\infty$-space ( = symmetric monoidal $\infty$-groupoid) on a single generator. That means that a symmetric monoidal functor $\mathcal F\mathrm{in}^\simeq \to \mathrm{Sp}$ (which always factors through the maximal subgroupoid $\mathrm{Sp}^\simeq\subseteq\mathrm{Sp}$) is equivalent to the data of a spectrum $M\in \mathrm{Sp}$ (the image of the singleton set). The functor is given by sending a finite set $I$ to the smash product $M^{\otimes I}$, and is evidently symmetric monoidal. The same Kan extension game as before now gives rise to an $S$-graded $\mathbb E_\infty$-ring spectrum, this time with underlying $\mathbb E_\infty$-ring
$$
\mathrm{Sym}^*(M)\simeq \bigoplus_{n\ge 0}M^{\otimes n}_{h\Sigma_n},
$$
the free $\mathbb E_\infty$-ring generated by $M$. We recover the prior situation by setting $M=S$. For $M = S^{\oplus n}$, we obtain an $S$-grading on $S\{t_1, \ldots, t_n\}$, corresponding to the spectral-algebro-geometric smooth affine $n$-space $\mathbf A^n$.
3. $S$-grading on $S\{t^{\pm 1}\}$
Another way to generalize the example of the $S$-grading on $S\{t\}$ is to take directly the constant functor $\Omega^\infty(S)\to \mathrm{Sp}$ with value $S$, instead of starting with a constant functor on $\mathcal F\mathrm{in}^\simeq$ and Kan-extending it. This produces a perfectly good $S$-graded $\mathbb E_\infty$-ring, which let us denote $S\{t^{\pm 1}\}$.
From the group-completion relationship between $\Omega^\infty(S)$ and $\mathcal F\mathrm{in}^\simeq$, it may be deduced that $S\{t^{\pm 1}\}$ and $S\{t\}$ are related in terms of $\mathbb E_\infty$-ring localization as $S\{t^{\pm 1}\}\simeq S\{t\}[t^{-1}]$, justifying our notation. Here we are localizing $S\{t\}$ at the element $t\in \mathbf Z[t] = \pi_0(S\{t\})$. In terms of spectral algebraic geometry, this is encoding the spectral scheme $\mathrm{GL}_1$, the smooth punctured line.
4. Remark on non-negative grading
In algebraic geometry, we often prefer to think about non-negatively graded commutative rings than graded commutative rings. Just as the latter are equivalent to lax symmetric monoidal functor $\mathbf Z\to\mathrm{Ab}$, so are the former equivalent to lax symmetric monoidal functors $\mathbf Z_{\ge 0}\to \mathrm{Ab}$.
By analogy, the "non-negatively $S$-graded $\mathbb E_\infty$-rings" are lax symmetric monoidal functors $\mathcal{F}\mathrm{in}^\simeq \to\mathrm{Sp}$. Indeed, just as $\mathbf Z_{\ge 0}$ is the free commutative monoid on one generator, so is $\mathcal F\mathrm{in}^\simeq$ the free $\mathbb E_\infty$-space on one generator. That is the reason why we were encountering such functors above (and Kan extending them along group completion, as we would to view as $\mathbf Z_{\ge 0}$-grading as a special case of a $\mathbf Z$-grading).
5. Some actual "non-tautological" examples though
So far, a not-completely-unreasonable complaint might be that all the examples of $S$-graded $\mathbb E_\infty$-rings were sort of tautological.
For a very non-tautological example, see the main result of this paper of Hadrian Heine. It shows that there exists an $S$-graded $\mathbb E_\infty$-ring spectrum such that its $S$-graded modules are equivalent to the $\infty$-category of cellular motivic spectra. In fact, much more is proved: this situation is very common, and under some not-too-harsh compact-dualizable-generation assumptions, a symmetric monoidal stable $\infty$-category will be equivalent to $S$-graded modules over some $S$-graded $\mathbb E_\infty$-ring. So you may just as well view this result as a wellspring of potentially interesting examples of $S$-gradings "occurring in nature"! :)
Best Answer
Interesting question! I can't give a real answer, but here are some idle musings:
Note that one way to encode exterior powers is as $\Lambda^i_R(E) = \Sigma^{-i}(\mathrm{Sym}^i_R(\Sigma(E)))$ (since placing the generators in degree one switches, by virtue of the Koszul sign rule from the usual $\Sigma_n$-action to the alternating one). So the symmetric algebra in SAG or DAG on a $1$-shifted module looks like a sheared version of the exterior algebra. That is, if $R$ is an ordinary commutative ring and $M$ an $R$-module, then $$ \mathrm{Sym}^*_R(\Sigma (M)) \simeq \Sigma^{*}(\Lambda_R^*(M)) $$ (though here both symmetric and exterior algebras are understood in the spectral or derived sense, and may not agree with their usual algebraic counterparts unless $M$ is flat). In the DAG context, we similarly have $$ \mathrm{Sym}^*_R(\Sigma^2(M)) \simeq \Sigma^{2*}(\Gamma_R^*(M)), $$ where $\Gamma^*_R(M)$ is the free divided power $R$-algebra generated by $M$. See Section 25.2 of the SAG book for the proofs.
Therefore, perhaps the candidates for your degree $n$ higher exterior algebras might be some kind of $n*$-fold desuspensions of the symmetric algebra $\mathrm{Sym}^*_R(\Sigma^n(R))$. For $n=0$, that would recover (flatness issues notwithstanding) the usual symmetric algebra, for $n=1$ the exterior algebra, for $n=2$ the free divided power algebra, and for $n\ge 3$, who knows? :)
Operating on this provisional understanding of what the higher exterior algebras are, let us discuss the $S$-grading. We see from the construction that the "unshearing" of it is the symmetric algebra $\mathrm{Sym}^*_R(\Sigma^n(R))$, which does indeed carry an $S$-grading. The question about whether the higher exterior algebras themselves carry a canonical $S$-grading now becomes a question about how, if at all, some kind of a "shearing" functor $(M_n)\mapsto (\Sigma^n M_n)$ interacts with $S$-gradings. I believe this going to be a very non-trivial question.
Even if we were not talking about the much-more-complicated $S$-gradings, but rather $\mathbf Z$-gradings, the shearing functor $\Sigma^*$ is indeed an automorphism of the $\infty$-category of $\mathbf Z$-graded spectra $\mathrm{Fun}(\mathbf Z,\mathrm{Sp})$, but I don't think it is symmetric monoidal. Indeed, consider rather its inverse squared $\Sigma^{-2*}$. If it were symmetric monoidal, it would send the polynomial $\mathbb E_\infty$-algebra $S[t]\simeq \bigoplus_{n\ge 0}S$, with its usual $\mathbf Z$-grading, into the "$2$-shifted polynomial $\mathbb E_\infty$-algebra" $S[\beta] \simeq \bigoplus_{n\ge 0}\Sigma^{-2n}S$. The latter thing does indeed exist as an $\mathbb E_2$-ring, studied in Section 2.8 of this paper by Lurie. But I have been told before that, due to some obstructions one can calculate, it can be shown that said $\mathbb E_2$-ring does not support an $\mathbb E_\infty$-structure. That would seem to imply that the shearing functor $\Sigma^{-2*}$ (and consequently $\Sigma^*$) can not be symmetric monoidal.
And that was just in the $\mathbf Z$-graded world, let alone all the additional complications that the $S$-graded setting might bring!