We present radial velocity observations of four extremely low-mass (0.2 M sun) white dwarfs (WDs). All four stars show peak-to-peak radial velocity variations of 540-710 km s-1 with 1.0-5.9 hr periods. The optical photometry rules out main-sequence companions. In addition, no millisecond pulsar companions are detected in radio observations. Thus, the invisible companions are most likely WDs. Two of the systems are the shortest period binary WDs yet discovered. Due to the loss of angular momentum through gravitational radiation, three of the systems will merge within 500 Myr. The remaining system will merge within a Hubble time. The mass functions for three of the systems imply companions more massive than 0.46 M sun; thus, those are carbon/oxygen core WDs. The unknown inclination angles prohibit a definitive conclusion about the future of these systems. However, the chance of a supernova Ia event is only 1%-5%. These systems are likely to form single R Coronae Borealis stars, providing evidence for a WD + WD merger mechanism for these unusual objects. One of the systems, SDSS J105353.89+520031.0, has a 70% chance of having a low-mass WD companion. This system will probably form a single helium-enriched subdwarf O star. All four WD systems have unusual mass ratios of <=0.2-0.8 that may also lead to the formation of AM CVn systems. Based on observations obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona.