  Almost all quantum Hall effect to date can be understood as {\em integral}
quantum Hall effect of appropriate particles, namely electrons or composite
fermions. This paper investigates theoretically the feasibility of nested
states of composite fermions which would lead to a quantum Hall effect that
cannot be understood as integral quantum Hall effect of composite fermions. The
weak residual interaction between composite fermions will play a crucial role
in the establishment of such quantum Hall states by opening a gap in a
partially filled composite-fermion level. To treat the problem of interacting
composite fermions, we develop a powerful method that allows us to obtain the
low energy spectra at composite fermion fillings of $\nu^*=n+\bar \nu$ without
making any assumption regarding the structure of composite fermions in the
topmost partially filled level. The method is exact aside from neglecting the
composite-fermion Landau level mixing, and enables us to study rather large
systems, for example, 24 particles at a total flux of 62 $hc/e$, for which the
dimension of the lowest Landau level Hilbert space is $\sim 10^{17}$. We have
investigated, for fully spin polarized composite fermions, several filling
factors between 1/3 and 2/5 using this approach. The results indicate that any
possible incompressibility at these fractions is likely to have a fundamentally
different origin than that considered earlier.
