It is well-known that if a based, path-connected topological space (X,e) is well-pointed, then the fundamental group of the James reduced product J(X) is naturally isomorphic to the first singular homology group H1(X). In this paper, we study the group π1(J(X),e) in much greater generality. If (X,e) is not well-pointed, then π1(J(X),e) may be a proper quotient of H1(X), which behaves in many ways as an infinitary abelianization of π1(X,e) for infinite products formed at the basepoint e. Our main result is that the canonical injection σ:X→J(X) induces a surjection σ#:π1(X,e)→π1(J(X),e) on fundamental groups for any path-connected Hausdorff space X. Equipped with this result and previous work on infinite commutativity in fundamental groups of monoids, we characterize the group π1(J(X),e) for a variety of examples.