We report on four radio-detected cosmic-ray (CR) or CR-like events observed with the Antarctic Impulsive Transient Antenna (ANITA), a NASA-sponsored long-duration balloon payload. Two of the four were previously identified as stratospheric CR air showers during the ANITA-I flight. A third stratospheric CR was detected during the ANITA-II flight. Here, we report on characteristics of these three unusual CR events, which develop nearly horizontally, 20-30 km above the surface of Earth. In addition, we report on a fourth steeply upward-pointing ANITA-I CR-like radio event which has characteristics consistent with a primary that emerged from the surface of the ice. This suggests a possible τ-lepton decay as the origin of this event, but such an interpretation would require significant suppression of the standard model τ-neutrino cross section.