In an earlier paper, we showed that the causal boundary of any homogeneous plane wave satisfying the weak energy condition consists of a single null line. For conformally flat plane waves such as the Penrose limit of AdS_5 x S^5, all spacelike curves that reach infinity also end on this boundary and the completion is Hausdorff. However, the more generic case (including, e.g., the Penrose limits of AdS_4 x S^7 and AdS_7 x S^4) is more complicated. In one natural topology, not all spacelike curves have limit points in the causal completion, indicating the need to introduce additional points at `spacelike infinity'--the endpoints of spacelike curves. We classify the distinct ways in which spacelike curves can approach infinity, finding a two-dimensional set of distinct limits. The dimensionality of the set of points at spacelike infinity is not, however, fixed from this argument. In an alternative topology, the causal completion is already compact, but the completion is non-Hausdorff.