Fundamental groups of topological groups.

Here is an example: a product of infinitely many $\mathbb{RP}^\infty$'s.

The crucial thing thing to see is that $\mathbb{RP}^\infty$ (or, easier to see, its universal cover $S^\infty$) has a group structure whose underlying group is a vector space of dimension $2^{\aleph_0}$. This is not hard: the total space $S^\infty$ of the universal $\mathbb{Z}_2$-bundle is obtained by applying a composite of functors to the group structure $\mathbb{Z}_2$ in the category of sets:

$$\textbf{Set} \stackrel{K}{\to} \textbf{Cat} \stackrel{\text{nerve}}{\to} \textbf{Set}^{\Delta^{op}} \stackrel{R}{\to} \textbf{CGHaus}$$

($\textbf{CGHaus}$ here is the category of compactly generated Hausdorff spaces and continuous maps). Here $K$ is the right adjoint to the "underlying set of objects" functor; it takes a set to the category whose objects are the elements of the set and there is exactly one morphism between any two objects. The functor $R$ is of course geometric realization.

Each of these functors is product-preserving, and since the concept of group can be formulated in any category with finite products, a product-preserving functor will map a group object in the domain category to one in the codomain category. Even more: the concept of a $\mathbb{F}_2$-vector space makes sense in any category with finite products since we merely need to add the equation $\forall_x x^2 = 1$ to the axioms for groups, which can be expressed by a simple commutative diagram.

Thus $S^\infty$ is an internal vector space over $\mathbb{F}_2$ in $\textbf{CGHaus}$. It can also be considered an internal vector space over $\mathbb{F}_2$ in $\textbf{Top}$, the category of ordinary topological spaces, because a finite power $X^n$ in $\textbf{Top}$ of a CW-complex $X$ has the same topology as $X^n$ does in $\textbf{CGHaus}$ provided that $X$ has only countably many cells, which is certainly the case for $S^\infty$ (see Hatcher's book, Theorem A.6). Thus $S^\infty$ can be considered as an honest commutative topological group of exponent 2.

The underlying group of $S^\infty$ (in $\textbf{Set}$) is clearly a vector space of dimension $2^{\aleph_0}$. We make take this vector space to be the countable product $\mathbb{Z}_2^{\mathbb{N}}$. Modding out by $\mathbb{Z}_2$ (modding out by a 1-dimensional subspace), the space $\mathbb{RP}^\infty$ is also, as an abstract group, isomorphic to this. And so is a countably infinite product $(\mathbb{RP}^\infty)^{\mathbb{N}}$ of copies of $\mathbb{RP}^\infty$.

Finally, the functor $\pi_1$ is product-preserving, and so

$$\pi_1((\mathbb{RP}^\infty)^{\mathbb{N}}) \cong \mathbb{Z}_{2}^{\mathbb{N}}$$

and we are done.

Here is another construction, which I think shows that it isn't surprising that such groups exist. First, we note the fact that for any abelian group $G$, there is a model of $BG$ that is an abelian topological group (there are many ways to see this; Todd's answer mentions one). Now start with any nontrivial abelian group $G_0$, and let $G_1$ be the underlying (discrete) group of $BG_0$. Similarly, let $G_2$ be the underlying group of $BG_1$, and so on. Now let $G=G_0\times\prod BG_n$. Then $\pi_1(G)=\prod G_n$ is isomorphic to $G$ as an abstract group. If you start with $G_0$ that is not just a discrete group but a simply connected topological group (e.g., $\mathbb{R}$), this gives you an example where $G$ is connected.

(Technically, this construction may require you to work in CGHaus rather than Top--it works in any category of spaces in which for any abelian $G$, you can construct an abelian group object whose fundamental group is $G$. I don't know if this is possible in Top.)

This question occurred as Advanced Problem 5889 in the Amer. Math. Monthly 80 (1973), no. 1, 82.  It was listed as still unsolved five years later, in vol. 85, no. 10, p. 834, of the Monthly; however, my recollection is that it mysteriously vanished from the Monthly's "unsolved" list the next time this got updated, but without a solution having appeared in the interim.