  We introduce a family of cluster algebras of infinite rank associated with
root systems of type $A$, $D$, $E$. We show that suitable completions of these
cluster algebras are isomorphic to the Grothendieck rings of the categories
$\mathcal{O}_\mathbb{Z}$ of the corresponding shifted quantum affine algebras.
The cluster variables of a class of distinguished initial seeds are certain
formal power series defined by E. Frenkel and the second author, which satisfy
a system of functional relations called $QQ$-system. We conjecture that all
cluster monomials are classes of simple objects of $\mathcal{O}_\mathbb{Z}$. In
the final section, we show that these cluster algebras contain infinitely many
cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat
cell of the corresponding simple simply-connected algebraic group. This
explains the similarity between $QQ$-system relations and certain generalized
minor identities discovered by Fomin and Zelevinsky.
