Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin [http://www.math.lsa.umich.edu/~ablass/bb.pdf][1] prove, among many other results, that $\mathbb{Z}^{\omega}$ (the infinite abelian group of integer-valued sequences under addition) is never the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$, if the chain has length less than $cov(B)$. They note it might be possible to remove the hypothesis on the length of the chain. However, they raise the **question** whether e.g. under $CH$, $\mathbb{Z}^{\omega}$ could be the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$. If the chain $\lbrace A_{\alpha} : \alpha < \delta \rbrace$ has the additional property that $\mathbb{Z}^{\omega}/A_{\alpha}$ is cotorsion-free for all $\alpha$, the answer is negative.

Has there been any further progress on this question?

One could replace $\omega$ by any countable infinite index set $A$, for example $\mathbb{Q}$, if this might help. Section 6 of George Bergman's paper exploits this possibility to define unusual subgroups of $\mathbb{Z}^{\omega}$: [http://math.berkeley.edu/~gbergman/papers/free_dual.pdf][2] 

  [1]: http://www.math.lsa.umich.edu/~ablass/bb.pdf
  [2]: http://math.berkeley.edu/~gbergman/papers/free_dual.pdf