Polynomials over a non-abelian group might look like $f(x) = x^7ax^3bxc$. Directly moving from field to group, we would define the usual degree as the sum of the exponents of the variable $x$.
Digression: This might be useful for the analogy, but if you wanted to compute (using a machine) the value a large polynomial in a group, you would probably want to get rid of exponential notation and use only the group law and grouping $(\cdot)$, e.g. $f(x) = (x\cdot x)^3x a x^3 b xc$, where for economy of notation we allow exponents only up to $3$, as this will still directly given a recipe for a computer to follow that is minimal in number of group operations.
So let's keep degree analogous to field theory of polynomials.
Clearly, for some $f(x)$, the equation $f(x) = 1$ has a solution, and in other cases it does not, just as in field theory. Anyone try this? Intuitively, it seems like it would have far reaching applications since the theory of field extensions does.
Please comment, or give a reason why this is fruitless. I don't care if nobody now cares about this, nor do I have time to explore in depth every idea, since no one is paying me to, and I eat food and have to buy things to survive.
In the category of groups $\mathbb{Z}$ has the nice property that for any $H$ and $h \in H$ there is a unique morphism $\mathbb{Z} \to H, 1 \mapsto h$. As the coproduct of $G$ and $\mathbb{Z}$ the generalized polynomial group satisfies the property that for any $G \to H$ and $h \in H$ there is a unique $G[x] \to H$ such that $x \mapsto h$ and $G \to G[x] \to H$ commutes. So it actually satisfies the same universal property that $A \to A[x]$ does in the category of rings. However this is where the similarity abruptly starts to end.
For an inclusion $G \hookrightarrow H$ we do have some notion of minimality, namely the least 'degree' of a polynomial over $G$ such that $h$ is a root. Unfortunately I doubt that a polynomial with least degree is unique in any meaningful sense. We also certainly do not have many of the nice properties like the fact that the minimal polynomial 'divides' or is part of the word of any other polynomial of which $h$ is a root. Alot of this comes from the fact that we don't have an appropriate notion of $0$ in a group. Sure we can solve for $p(h) = e$, but $p(h)q(h) \neq e$ in general and so on
I'm sure that study of these objects will yield some results, but they won't be as elementary or as powerful results from polynomials over fields.
This isn't a very good answer and I'm not versed enough in group theory to provide any facts about group polynomials. However if you still have interest in these objects I encourage you to study them. Maybe this answer and others will help you find some references.