We consider a natural algorithm for producing a random spanning tree for the Cayley graph of a finitely generated, countably infinite group. Associated with such a tree there is a (deterministic) compactification of the group: a sequence of group elements that is not eventually constant is convergent if the random geodesic through the spanning tree that joins the identity to the nth element of the sequence converges in distribution as n goes to infinity. We identify this compactification in a number of simple examples and present some open problems.