This paper by P.-E. Caprace, uses a "refined boundary" of a CAT(0) space. This boundary is constructed in the following way : given a point $\xi$ in the boundary at infinity of your space $X$, you construct a point $X_\xi$, which is the inverse limit of the horoballs centered at $\xi$. Here the maps $\phi_{m,n}$ are the CAT(0) projections. Then the space $X_\xi$ is itself CAT(0), and you can iterate the construction. Under reasonable hypotheses, the construction stops after a finite number of steps, and the refined boundary is the union of all the spaces you get.

(In the case of symmetric spaces, this construction has been already considered by Karpelevic in 1965, but with different definitions, and I don't think he saw it as inverse limits).