Fundamental group of Hawaiian earring

I am trying to understand how the fundamental group of the infinite shrinking wedge of circles is $G=\prod_{i=1}^\infty\mathbb{Z}$.

I understand that it is something more than $H=\bigoplus_{i=1}^\infty\mathbb{Z}$ because we can get more loops than $H$ admits because the radii of circles decrease, and continuity only requires we approach $0$ rather than terminate at $0$.

However, I feel like the fundamental group is still something more than $G$. Namely, we should be able visit previous circles with larger radii as long as these visits only occur a finite number of times (this groups operation is not pointwise multiplication but rather alternating letter weaving where consecutive letters from the same copy of $\mathbb{Z}$ are added together).

It could be that these two groups I'm considering are isomorphic, but I have no clue how to show that or if they even are. Thus my question is are these two groups isomorphic? or does the group I'm considering not represent the fundamental group of the infinite shrinking wedge of circles?

EDIT

I'm embarrassed to say that I've read an assertion that wasn't made in my source--- It merely says that the fundamental group of the wedge surjects onto $G$. Thus I'm asking if anyone does know the fundamental group of the infinite shrinking wedge?

-
I don't think the fundamental group of the infinitely shrinking wedge of circles is an infinite direct product of $\mathbb{Z}$. I am not sure where you get your source of information, but I found something similar on Hatcher, which introduce a homomorphism from the fundamental group to G. Then he says, the maps is surjective, but not injective. "The fundamental group is actually far more complicated than G" since it is non-abelian –  Tian Feb 7 at 6:12

The fundamental group of the infinite shrinking wedge of circles is not equal to $G=\prod_{i=1}^\infty\mathbb{Z}$. Hatcher only makes the claim that $\pi_1(X)$ surjects onto $G$ and that $\pi_1(X)$ is therefore uncountable. The fundamental groups of $X$ is definitely larger than $G$, but there is probably no nice representation of it.
In this article, one gives a description of $\pi_1(H)$ as a subgroup of the projective limit of the free groups on a finite set of generators. (In particular, the article aims to prove that $\pi_1(H)$ isn't free.)