Little is known about the origin of the fastest stars in the Galaxy. Our understanding of the chemical evolution history of the Milky Way and surrounding dwarf galaxies allows us to use the chemical composition of a star to investigate its origin and to say whether it was formed in situ or was accreted. However, the fastest stars, the hypervelocity stars, are young and massive and their chemical composition has not yet been analyzed. Though it is difficult to analyze the chemical composition of a massive young star, we are well versed in the analysis of late-type stars. We have used high-resolution ARCES/3.5m Apache Point Observatory, MIKE/Magellan spectra to study the chemical details of 15 late-type hypervelocity star candidates. With Gaia EDR3 astrometry and spectroscopically determined radial velocities we found total velocities with a range of 274-520km/s and mean value of 381km/s. Therefore, our sample stars are not fast enough to be classified as hypervelocity stars, and are what is known as extreme-velocity stars. Our sample has a wide iron abundance range of -2.5<~[Fe/H]<~-0.9. Their chemistry indicates that at least 50% of them are accreted extragalactic stars, with iron-peak elements consistent with prior enrichment by sub- Chandrasekhar mass Type Ia supernovae. Without indication of binary companions, their chemical abundances and orbital parameters indicate that they are the accelerated tidal debris of disrupted dwarf galaxies.