What is the size of codomain of a function $G(x) = F^A(x) \oplus F^B(x)$, where $F(x) = \text{Keccak-}f[1600](x)$?
Let $\pi$ and $\sigma$ be two independent uniform random permutations, and $f$ a uniform random function.  The best advantage of any $q$-query algorithm to distinguish $\pi + \sigma$ from $f$ is bounded by $(q/2^n)^{1.5}$[1].  In this case, the expected fraction of distinct outputs of $\pi + \sigma$ can't be too far from the expected fraction of distinct outputs from $f$, which is $1 - e^{-1} \approx 63\%$.

What about $\sigma = \pi^2$, or $\sigma = \pi^k$ for $k &gt; 2$?  Then $\pi$ and $\sigma$ are not independent.  Nevertheless, it would be rather surprising if this situation were substantially different.

What about $\pi^{2^{3456789}} + \pi^{2^{987654321}}$ instead of $\pi + \pi^2$?  This is the same as $\pi + \pi^{2^{987654321 - 3456789}}$.  It's not clear why you would be worried about uncomputably large exponents like this unless you were flailing around without principle trying to make a design that looks complicated.