Examples of Abelian groups with certain non-terminating descreasing chains Q1. What are some examples of Abelian groups containing
 a chain of strictly decreasing subgroups $\{G_n\}_{n\in\mathbb Z} $  (i.e. $G_{n+1}<G_n$ and $G_{n+1}\not=G_n$ ) such that
i)$\cup_n G_n=G \text{ and } \cap_n G_n=1$ (non-Artinian covering )
ii)
$\sup\{\text{order}(\frac{G_n}{G_{n+1}})\: n\in\mathbb Z\}<\infty$ (order of quotient groups are controlled).
Q2. In particular, does additive group of reals $(\mathbb R,+)$ contain such a chain? 
 A: For each $k\in {\bf Z}$, choose your favourite finite abelian group $G_k$ of order at most $N$ (where $N$ is your favourite natural number greater than $1$). Then the set of all sequences $(g_k)_k$ with $g_k\in G_k$ and all but finitely many $g_k= e_{G_k}$ is an example.
Alternately, take all sequences such that eventually $g_k=e_{G_k}$.
Moreover, you can take any intermediate groups between the two.
A: There are many examples.  For a particularly simple one, let $(A_k)_{k\in\mathbb{Z}}$ be any sequence of nontrivial abelian groups of bounded order, let $G=\bigoplus_{k\in\mathbb{Z}}A_k$, and let $G_n=\bigoplus_{k\geq n}A_k$.  Or you could let $G$ be the subgroup of $\prod_{k\in\mathbb{Z}}A_k$ consisting of elements whose support is bounded below, and $G_n=\prod_{k\geq n} A_k$.
For another sort of example, let $p$ be a prime number and take $G=\mathbb{Z}[1/p]$, with $G_n=p^n\mathbb{Z}$.  Variations on this example include $G=\mathbb{Q}$ with $G_n=p^n\mathbb{Z}_{(p)}$, or $G=\mathbb{Q}_p$ with $G_n=p^n\mathbb{Z}_p$.  Since $\mathbb{Q}_p$ is isomorphic to $\mathbb{R}$ as a group (they are both just $\mathbb{Q}$-vector spaces of cardinality $2^{\aleph_0}$), this answers your second question.
