In the extreme quantum limit, when the Landau level filling factor ν<
1, the dominant electron-electron interaction in low-disorder two-dimensional electron systems leads to exotic many-body phases. The ground states at even-denominator ν=1/2 and 1/4 are typically Fermi seas of composite fermions carrying two and four flux quanta, surrounded by the Jain fractional quantum Hall states (FQHSs) at odd-denominator fillings ν=p/(2p±1) and ν=p/(4p±1), where p is an integer. For ν<
1/5, an insulating behavior, which is generally believed to signal the formation of a pinned Wigner crystal, is seen. Our experiments on ultra-high-quality, dilute, GaAs two-dimensional electron systems reveal developing FQHSs at ν=p/(6p±1) and ν=p/(8p±1), manifested by magnetoresistance minima superimposed on the insulating background. In stark contrast to ν=1/2 and 1/4, however, we observe a pronounced, sharp minimum in magnetoresistance at ν=1/6 and a somewhat weaker minimum at ν=1/8, suggesting developing FQHSs, likely stabilized by the pairing of composite fermions that carry six and eight flux quanta. Our results signal the unexpected entry, in ultra-high-quality samples, of FQHSs at even-denominator fillings 1/6 and 1/8, which are likely to harbor non-Abelian anyon excitations.