This question is motivated by (some available information on) this MO-problem on the largest possible degree of a polynomial on a finite group and this MO-problem on the degree of the constant polynomials on a finite group.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. The smallest number $n\in\mathbb N$ in such a representation of $f$ is called the degree of the polynomial $f$ and is denoted by $\deg(f)$.
Each polynomial $f$ on a commutative group $X$ is of the form $f(x)=ax^n$ for some $a\in X$ and $n\in\mathbb N$, which implies that
$\bullet$ any polynomial $f$ on the commutative group $X$ has degree $\deg(f)\le\exp(X):=\min\{n\in\mathbb N:\forall x\in X\;\;(x^n=1)\}$,
$\bullet$ the constant polynomial $1_X:X\to \{1\}\subseteq X$ on $X$ has degree $\deg(1_X)=\exp(X)$, and
$\bullet$ the set $\mathrm{Poly}(X)$ of all polynomials on $X$ has cardinality $|\mathrm{Poly}(X)|=|X|\cdot\exp(X)$.
By Theorem in this MO-post, a finite group $X$ has $|\mathrm{Poly}(X)|=|X|\cdot\exp(X)$ if and only if $X$ is commutative or $X$ is isomorphic to $Q_8\times A$ for some nontrivial Abelian group $A$ of odd order.
What about the other two properties of commutative groups:
Problem 1. Characterize finite groups $X$ such that $\deg(1_X)=\exp(X)=\max\{\deg(f):f\in\mathrm{Poly}(X)\}.$
Remark 1. GAP-calculations show that there exist at least 7 finite groups $X$ with $$\deg(1_X)=\exp(X)=\max\{\deg(f):f\in Poly(X)\}=4.$$ Those are the groups $D_8$, $Q_8$, $(C_4\times C_2):C_2$, $C_4:C_4$, $C_2\times D_8$, $C_2\times Q_8$, $(C_4\times C_2):C_2$ (indexed by (8,3), (8,4), (16,3), (16,4), (16,11), (16,12), (16,13) in GAP).
Remark 2. For any nontrivial finite Abelian group $A$ of odd order, the group $X=Q_8\times A$ has $\deg(1_X)=\max\{\deg(f):f\in\mathrm{Poly}(X)\}=\exp(X)=4\exp(A)$ and $|\mathrm{Poly}(X)|=|X|\cdot\exp(X)$, so there are infinitely many non-commutative groups with the property from Problem 1.
Remark 3. Answering Question 2 in this MO-post, Peter Taylor observed that the groups $A_4$ and $C_3\times S_3$ are the unique groups $X$ of cardinality $|X|<24$ with $\max\{\deg(f):f\in\mathrm{Poly}(X)\}>\exp(X)$. GAP-calculations also show that these two groups are the unique non-commutative groups $X$ of order $|X|<20$ such that $\deg(1_X)>4$. More precisely,
$\bullet$ $6=\exp(A_4)=\deg(1_{A_4})<8=\max\{\deg(f):f\in\mathrm{Poly}(A_4)\}$, and
$\bullet$ $6=\exp(C_3\times S_3)=\deg(1_{C_3\times S_3})<9=\max\{\deg(f):f\in\mathrm{Poly}(C_3\times S_3)\}$.
Remark 4. It is known that each non-commutative finite simple group $X$ has $\exp(X)\le \frac12|X|$. On the other hand, a non-commutative finite group $X$ is simple if and only if $\mathrm{Poly}(X)=X^X$, see this MO-post. By the comment of Emil Jeřábek to this MO-post, the equality $\mathrm{Poly}(X)=X^X$ implies that $$\max\{\deg(f):f\in\mathrm{Poly}(X)\}\ge |X|-1>\frac12|X|\ge\exp(X).$$ So, non-commutative finite simple groups do not have the property from Problem 1. However, at the moment it is not clear whether $\deg(1_X)<\exp(X)$ for any non-commutative finite simple group $X$. Noam D. Elkies has shown that $\deg(1_{A_5})=10<30=\exp(A_5)$.
