Yes, there exists a natural Hopf algebra structure on the ring of symmetric functions (i. e., symmetric "polynomials" in infinitely many indeterminates). It is not related to the additive group of $k^n$ (for a good reason: as you said, the obvious coalgebra structure on $k\left[x_1,x_2,...,x_n\right]$ coming from addition on the affine space does not survive restriction to the symmetric polynomials), and it does require infinitely many indeterminates.
1. The algebra $\mathbf{Symm}$
Let $k$ be a commutative ring. (It will serve as a base ring. You can safely assume $k$ to be $\mathbb C$ if you want, but if you know some representation theory, you can actually get surplus value from working with $k=\mathbb Z$ even if all you care about are representations of finite groups over $\mathbb C$.)
So what is $\mathbf{Symm}$ (or $Sym$, as it is also known)? First of all, it is a $k$-Hopf algebra which, as a $k$-algebra, is the algebra of "symmetric functions" in countably many commuting indeterminates $X_1$, $X_2$, $X_3$, .... Here, "function" does not mean an actual function, but instead it means a power series $p$ in the indeterminates $X_1$, $X_2$, $X_3$, ... such that for some $d\in \mathbb N$, every monomial of total degree $\geq d$ occurs in $p$ with coefficient $0$. (Another word for this meaning of "function" is "degree-bounded power series".) "Symmetric" means that any two monomials which only differ in the order of the exponents (but the multisets of their exponents are the same) must have the same coefficient. (I guess I should say that for me, a monomial doesn't include a coefficient.) This definition of $\mathbf{Symm}$ is easily seen to be equivalent to the definition as a direct limit $\lim\limits_{n\to\infty} k\left[X_1,X_2,...,X_n\right]^{S_n}$, where the mapping $k\left[X_1,X_2,...,X_n\right]^{S_n} \to k\left[X_1,X_2,...,X_m\right]^{S_m}$ (for $n\leq m$) takes a symmetric polynomial in $n$ variables and clones its coefficients to get an $m$-variable symmetric polynomial (I hope this is understandable; anyway, this is not important for us).
2. Applying symmetric functions to multisets
Since we call the elements of $\mathbf{Symm}$ "functions", let us explain what they can evaluated at:
For any $p\in\mathbf{Symm}$, any $u\in\mathbb N$ and any $u$-tuple $\left(x_1,x_2,...,x_u\right)$ of elements of a $k$-algebra, we can define the "value" of $p$ at $\left(x_1,x_2,...,x_u\right)$ (denoted by $p\left(x_1,x_2,...,x_u\right)$) to be the result of
1) removing all monomials which contain at least one of the variables $X_{u+1}$, $X_{u+2}$, $X_{u+3}$, ... from the power series $p$;
2) then applying the resulting power series $q$ to $X_1=x_1$, $X_2=x_2$, ..., $X_u=x_u$ (this makes sense because the power series $q$ is a polynomial, due to our definition of "function").
Due to the symmetry of $p$, this result actually doesn't depend on the order of $\left(x_1,x_2,...,x_u\right)$; thus, we can think of $p\left(x_1,x_2,...,x_u\right)$ as the value of $p$ at the multiset (rather than $u$-tuple) $\left(x_1,x_2,...,x_u\right)$.
3. The bialgebra $\mathbf{Symm}$
So we have a $k$-algebra $\mathbf{Symm}$. How do we make it a Hopf algebra? First, let me show the intuition behind the comultiplication. It should correspond to the union of multisets. So, if we have some $p\in\mathbf{Symm}$, and write $\Delta\left(p\right)=\sum\limits_{i \in I} q_i \otimes r_i$, then we should have
(1) $p\left(x_1,x_2,...,x_u,y_1,y_2,...,y_v\right) = \sum\limits_{i \in I} q_i\left(x_1,x_2,...,x_u\right) r_i\left(y_1,y_2,...,y_v\right)$ for any two multisets $\left(x_1,x_2,...,x_u\right)$ and $\left(y_1,y_2,...,y_v\right)$ of elements of any $k$-algebra.
The counity should mean applying to the empty multiset:
(2) $p\left(\ \right) = \varepsilon\left(p\right)$.
(Sorry, dear Java friends, $p\left(\ \right)$ doesn't mean $p$ here.)
How do we actually get such $\Delta$ and $\varepsilon$ ? The easy answer is: By the fundamental theorem on symmetric polynomials, $\mathbf{Symm}$ is generated by the $k$-algebra by the elementary symmetric polynomials
$e_1 = X_1 + X_2 + X_3 + X_4 + ... = \sum\limits_{i=1}^{\infty} X_i$;
$e_2 = X_1 X_2 + X_1 X_3 + ... + X_2 X_3 + X_2 X_4 + ... + X_3 X_4 + ... = \sum\limits_{1\leq i < j} X_i X_j$;
$e_3 = X_1 X_2 X_3 + X_1 X_2 X_4 + ... + X_1 X_3 X_4 + ... + X_2 X_3 X_4 + ... = \sum\limits_{1\leq i < j < k} X_i X_j X_k$;
...;
$e_j = \sum\limits_{1\leq i_1 < i_2 < ... < i_j} X_{i_1} X_{i_2} ... X_{i_j}$;
...,
and these $e_1$, $e_2$, $e_3$, ... are algebraically independent.
(This does not immediately follow from the fundamental theorem on symmetric polynomials, since the fundamental theorem is usually not formulated for infinitely many variables, but you can either apply the same argument (lexicographic induction) or use the direct-limit construction of $\mathbf{Symm}$ to conclude the infinite-variables case from the finite-variables here.)
Hence, in order to define a $k$-algebra homomorphism from $\mathbf{Symm}$ to another $k$-algebra (be it $\mathbf{Symm}\otimes \mathbf{Symm}$ or $k$ or anything else), it is enough to specify its values at the $e_j$ for $j=1,2,3,...$, and we have total freedom to do so. Since we want $\mathbf{Symm}$ to be a $k$-bialgebra, we must define $\Delta$ and $\varepsilon$ as $k$-algebra homomorphisms; so let us define $\Delta$ by requiring that
$\Delta\left(e_j\right) = \sum\limits_{m=0}^j e_m \otimes e_{j-m}$ for all $j\geq 1$,
and let us define $\varepsilon$ by requiring that
$\varepsilon\left(e_j\right) = 0$ for all $j\geq 1$.
(Actually, $\varepsilon$ just maps every $p\in\mathbf{Symm}$ to the constant term of $p$. But $\Delta$ isn't that easily described.)
To check that this matches our intuition above at least on the $e_j$ (i. e., that it satisfies (1) and (2) for $p=e_j$), we must show that every $j\geq 1$ satisfies
$e_j\left(x_1,x_2,...,x_u,y_1,y_2,...,y_v\right) = \sum\limits_{m=0}^j e_m\left(x_1,x_2,...,x_u\right) e_{j-m}\left(y_1,y_2,...,y_v\right)$ for any two multisets $\left(x_1,x_2,...,x_u\right)$ and $\left(y_1,y_2,...,y_v\right)$ of elements of any $k$-algebra,
and $e_j\left(\ \right) = 0$. These are very easy. It is more complicated to check (1) and (2) for general $p$.
4. The graded Hopf algebra $\mathbf{Symm}$
From here on, everything goes very fast: We have a $k$-bialgebra $\mathbf{Symm}$. It is graded (the $n$-th degree consists of homogeneous power series of degree $n$ in $\mathbf{Symm}$) and connected (the $0$-th degree is the ground ring $k$, embedded in $\mathbf{Symm}$ by constant power series), so it is a Hopf algebra. This is because every connected graded bialgebra is a Hopf algebra (the antipode is constructed by induction on the degree); for a more detailed proof of this, see, e. g., Corollary II.3.2 in Dominique Manchon's http://arxiv.org/abs/math/0408405 , which even proves this for connected filtered bialgebras).
For an overview of the properties of $\mathbf{Symm}$, see, e. g. Michiel Hazewinkel's http://arxiv.org/abs/0804.3888 Section 10 (errata: http://www.cip.ifi.lmu.de/~grinberg/algebra/typos1short.pdf ) and see also his Section 18 about its relation to the representation theory of symmetric groups. The antipode, for example, switches elementary symmetric with complete symmetric functions (up to sign), leaving the power sum functions invariant (again, up to sign).
Several "extensions" of $\mathbf{Symm}$ are now well-known: $\mathbf{QSymm}$ and $\mathbf{NSymm}$ are discusssed in Hazewinkel's http://arxiv.org/abs/0804.3888 , http://arxiv.org/abs/math/0410468 and http://arxiv.org/abs/math/0410470 . The "further-up" extensions are newer: The Loday-Ronco Hopf algebra of trees (see Marcelo Aguiar, http://www.math.tamu.edu/~maguiar/Loday.pdf ) and the Malvenuto-Reutenauer Hopf algebra of permutations (see Marcelo Aguiar, http://www.math.tamu.edu/~maguiar/MR.pdf ). These things are usually studied over $k=\mathbb Z$, but everything you do over $\mathbb Z$ trivially extends to all other $k$'s.
