Number of n-th roots of elements in a finite group and higher Frobenius-Schur indicators
If $n > 2$, there is NO absolute upper bound on the "higher" F.S. indicator $s_n(\chi)$. This is Problem 4.9 in my character theory book. (A hint is given there.)
Here are some things you probably know. For a representation $W$ of $G$, let $\text{Inv}(W)$ denote the subspace of $G$-invariants. For an irreducible representation $V$ with character $\chi$, the F-S indicator $s_2(\chi)$ naturally appears in the formulas
$$\dim \text{Inv}(S^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 + \chi(g^2)}{2}$$
and
$$\dim \text{Inv}(\Lambda^2(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^2 - \chi(g^2)}{2}.$$
More precisely the F-S indicator is their difference, while their sum is $1$ if $V$ is self-dual and $0$ otherwise. The corresponding formulas involving $s_3(\chi)$ are
$$\dim \text{Inv}(S^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 + 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}$$
and
$$\dim \text{Inv}(\Lambda^3(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{\chi(g)^3 - 3 \chi(g^2) \chi(g) + 2 \chi(g^3)}{6}.$$
Here the F-S indicator $s_3(\chi)$ naturally appears in the sum, not the difference, of these two dimensions. $T^3(V)$ decomposes into three pieces, and the third piece is (Edit, 9/26/20: two copies of) the Schur functor $S^{(2,1)}(V)$, which therefore satisfies
$$\dim \text{Inv}(S^{(2,1)}(V)) = \frac{1}{|G|} \sum_{g \in G} \frac{ \chi(g)^3 - \chi(g^3)}{3}.$$
So $s_3(\chi)$ constrains the dimensions of these spaces in some more mysterious way than $s_2(\chi)$ does. The sum
$$\dim \text{Inv}(T^3(V)) = \frac{1}{|G|} \sum_{g \in G} \chi(g)^3$$
tell us whether $V$ admits a "self-triality," and this dimension is an upper bound on $s_3(\chi)$. If $V$ is self-dual, this is equivalent to asking whether there is an equivariant bilinear map $V \times V \to V$, which might be of interest to somebody. If this dimension is nonzero then $s_3(\chi)$ gives us information about how a triality behaves under permutation.
The situation for higher values of $3$ is worse in the sense that the bulk of the corresponding formulas are not completely in terms of F-S indicators but in terms of inner products of F-S indicators and their interpretation will only get more confusing. Already I don't know of many applications of triality (in fact I know exactly one: http://math.ucr.edu/home/baez/octonions/node7.html).
The concept of higher Frobenius-Schur indicators has been generalized to far broader contexts than just group representations, and these have proven extremely useful.
Linchenko and Montgomery generalized the second indicators to semisimple Hopf algebras, and Kashina, Sommerhauser, and Zhu generalized higher indicators to the same. They were then defined for spherical fusion categories by Ng and Schauenburg, and have been used in problems ranging from classifications of low dimensional (quasi-)Hopf algebras; relating exponents and dimensions; obtaining new gauge invariants; and much more— the intro for this paper by Negron and Ng gives a pretty solid rundown on these generalizations and their uses. Some of the generalizations do not require semisimplicity, which for the most part means the invariants are only of the regular representation (rather than arbitrary fin. dim. reps). Searching for "frobenius-schur indicators" on the math arxiv should pull up quite a number of results.
When applied to the quasi-triangular Hopf algebras D(G), the Drinfeld double of the group G, you get a number of new group invariants which are, loosely speaking, based on how many roots (of a given element) are sent to roots (of the same element) under the action of each element under the regular representation. It then becomes quite interesting to investigate how this depends on the given element. Sometimes it only depends on the cyclic subgroup it generates (such as for all regular p-groups), sometimes it doesn't (such as for some irregular p-groups and the Monster group). Ultimately these can be tied into how the group is pieced together from its one-element centralizers, which if you know much about the representation theory of D(G) is quite sensible for an invariant of Rep(D(G)): representations are given by inducing representations from centralizers to the whole group.
Quite recently, Barter, Jones, and Tucker showed how these indicators effectively govern the structure of certain annular Temperley-Lieb-Jones modules in the fusion categories. Such objects are specified by rotation eigenvalues (which all have a predictable modulus, provided you know the twists), and the indicators govern the coefficients of the polynomials that dictate the multiplicities of these eigenvalues.