Let $k$ be a field of characteristic zero, and let $X = \mathbb{A}^n_k$ be affine $n$-space. Up to $k$-rational translation, we may assume that a map $e : \operatorname{Spec} k \to X$ is the inclusion of the origin.
Are there any possible algebraic group laws on $X$ whose identity is the origin besides the natural one? Equivalently, are there any possible Hopf algebra structures on $k[x_1,\ldots, x_n]$ besides the following? $$ \mu : x_i \mapsto 1 \otimes x_i + x_i \otimes 1 $$ $$ i : x_i \mapsto -x_i $$ Even in one dimension, the algebra required to prove this is somewhat involved: using characteristic not 2, one can see that $i$ must be negation. Then writing the comultiplication as $x \mapsto p(x_1, x_2) \in k[x_1, x_2]$, it is easy to see that $p(x_1, x_2) \in x_1 + x_2 + (x_1 x_2)$, but a complicated induction with associativity seems to be necessary to handle the higher order terms (and indeed, the nonvanishing of all binomial coefficients is the only way I can imagine that characteristic zero can enter the proof).
Is there a way to use some algebraic geometry to sidestep the algebra, or a better way to organize the algebraic argument?
I'm also happy to use transcendental methods as long as the argument applies to all characteristic zero fields. Note that by Galois descent it suffices to show the result for $k$ algebraically closed . . . but it may be too large to embed into $\mathbb{C}$. Perhaps one can write down a universal Hopf algebra on $X$ over some finitely generated $\mathbb{Q}$-scheme base, and then check on an algebraically closed fiber to reduce to $\mathbb{C}$).
Yes, and they're not even so exotic. If $\mathfrak{g}$ is a nilpotent Lie algebra over a field of characteristic zero then the Baker-Campbell-Hausdorff formula terminates so it defines a polynomial group law on $\mathfrak{g}$ regarded as an affine space, which over $\mathbb{R}$ or $\mathbb{C}$ produces the corresponding simply connected nilpotent Lie group. For a more explicit example you can consider the $3$-dimensional Heisenberg group
$$\left[ \begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{array} \right]$$
whose multiplication defines a nonabelian polynomial group law on $\mathbb{A}^3$ (over any field), namely
$$(x_1, y_1, z_1) \ast (x_2, y_2, z_2) = (x_1 + x_2, y_1 + y_2, z_1 + z_2 + x_1 y_2).$$
This comes from applying the above construction to the $3$-dimensional Heisenberg Lie algebra (up to some minor change of coordinates); that $x_1 y_2$ term corresponds to the only nontrivial bracket in the Lie algebra. Alternatively this construction can be described as a central extension of $\mathbb{G}_a^2$ by $\mathbb{G}_a$ in the category of affine group schemes.
I'd guess that every polynomial group law on $\mathbb{A}^n$ arises this way in characteristic zero (up to change of coordinates) but I don't know a proof.
If instead of group laws you'd asked for formal group laws, which corresponds to working at the formal completion of affine space at the origin, then examples can be constructed from any finite-dimensional Lie algebra (by using BCH), and in characteristic zero this is known to be an equivalence of categories.