We initiate the study of complexity classes ${A^B}$ where ${A}$ and ${B}$ are both ${TFNP}$ subclasses. For example, we consider complexity classes of the form ${PPP^{PPP}}$, ${PPAD^{PPA}}$, and ${PPA^{PLS}}$. We define complete problems for such classes, and show that they belong in ${TFNP}$. These definitions require some care, since unlike a class like ${PPA^{NP}}$, where the ${NP}$ oracle defines a function, in ${PPA^{PPP}}$, the oracle is for a search problem with many possible solutions. Intuitively, the definitions we introduce quantify over all possible instantiations of the ${PPP}$ oracle.

With these notions in place, we then show that several ${TFNP}$ subclasses are self-low. In particular, ${PPA^{PPA}} = {PPA}$, ${PLS^{PLS}} = {PLS}$, and ${LOSSY^{LOSSY}} = {LOSSY}$. These ideas introduce a novel approach for classifying computational problems within ${TFNP}$, such as the problem of deterministically generating large primes.