  Holm and Jorgensen have shown the existence of a cluster structure on a
certain category $D$ that shares many properties with finite type $A$ cluster
categories and that can be fruitfully considered as an infinite analogue of
these. In this work we determine fully the combinatorics of this cluster
structure and show that these are the cluster combinatorics of cluster algebras
of infinite rank. That is, the clusters of these algebras contain infinitely
many variables, although one is only permitted to make finite sequences of
mutations.
  The cluster combinatorics of the category $D$ are described by triangulations
of an $\infty$-gon and we see that these have a natural correspondence with the
behaviour of Plucker coordinates in the coordinate ring of a doubly-infinite
Grassmannian, and hence the latter is where a concrete realization of these
cluster algebra structures may be found. We also give the quantum analogue of
these results, generalising work of the first author and Launois.
  An appendix by Michael Groechenig provides a construction of the coordinate
ring of interest here, generalizing the well-known scheme-theoretic
constructions for Grassmannians of finite-dimensional vector spaces.
