  The theory ZFC implies the scheme that for every cardinal $\delta$ we can
make $\delta$ many dependent choices over any definable relation without
terminal nodes. Friedman, the first author, and Kanovei constructed a model of
ZFC$^-$ (ZFC without power set) with largest cardinal $\omega$ in which this
principle fails for $\omega$ many choices. In this article we study failures of
dependent choice principles over ZFC$^-$ by considering the notion of big
proper classes. A proper class is said to be big if it surjects onto every
non-zero ordinal.
  We shall see that if one assumes the scheme of dependent choices of any
arbitrary set length then every proper class is indeed big. However, by
building on work of Zarach, we provide a general framework for separating
dependent choice schemes of various lengths by producing models of ZFC$^-$ with
proper classes that are not big. Using a similar idea, we then extend the
earlier result by producing a model of ZFC$^-$ in which there are unboundedly
many cardinals but the scheme of dependent choices of length $\omega$ still
fails.
  Finally, the second author has proven that a model of ZFC$^-$ cannot have a
non-trivial, cofinal, elementary self-embedding for which the von-Neumann
hierarchy exists up to its critical point. We answer a related question posed
by the second author by showing that the existence of such an embedding need
not imply the existence of any non-trivial fragment of the von-Neumann
hierarchy. In particular, that in such a situation $\mathcal{P}(\omega)$ can be
a proper class.
