  We associate, with every infinite word over a finite alphabet, an increasing
sequence of rooted finite graphs, which provide a discrete approximation of the
famous Sierpi\'nski carpet fractal. Each of these sequences converges, in the
Gromov-Hausdorff topology, to an infinite rooted graph. We give an explicit
description of the metric compactification of each of these limit graphs. In
particular, we are able to classify Busemann and non-Busemann points of the
metric boundary. It turns out that, with respect to the uniform Bernoulli
measure on the set of words indexing the graphs, for almost all the infinite
graphs, the boundary consists of four Busemann points and countably many
non-Busemann points.

Metric Geometry