What is the size of codomain of a function $G(x) = F^A(x) \oplus F^B(x)$, where $F(x) = \text{Keccak-}f[1600](x)$? Assuming that $m$ is a multiset of bitstrings where all bitstrings have the same length, let $D(m)$ denote the number of distinct elements in $m$. That is, $D(m)$ is equal to the dimension of $m$. For example, if $$m = \{00, 10, 11, 10, 11\},$$ then $D(m)=3$.  

Let $F(x) = \text{Keccak-}f[1600](x)$, the block permutation function of SHA-3 (for $64$-bit words). We can define the following notation: $$\begin{array}{l}
{F^0(x)} = x,\\
{F^1(x)} = F(x),\\
{F^2(x)} = F(F(x)),\\
{F^3(x)} = F(F(F(x))),\\
\ldots
\end{array}$$ 

Assuming that $A$ and $B$ are two different natural numbers greater than or equal to $0$, let $G_{A, B}(x)$ denote a function defined as $$G_{A, B}(x) = F^A(x) \oplus F^B(x),$$

where $x$ denotes a $1600$-bit input and $\oplus$ denotes an XOR operation.  

Assuming that $L = 2^{1600}$, let $S_i$ denote an $i$-th bitstring from a set of all possible $1600$-bit inputs:
$$\begin{array}{l}
S_1 = 0^{1600},\\
S_2 = 0^{1599}1,\\
\ldots,\\
S_{L-1} = 1^{1599}0,\\
S_L = 1^{1600}.\\
\end{array}$$   

Let $A$ and $B$ denote two arbitrarily large, but different natural numbers (one of them is allowed to be equal to $0$). For example, $$A = 0, B = 1$$ or $$A = 2^{3456789}, B = 9^{876543210}$$ are valid pairs.  

Then    

$$\begin{array}{l}
S_{A, B}[i] = G_{A, B}(S_i),\\
C_{A, B} = \{S_{A, B}[1], S_{A, B}[2], \ldots, S_{A, B}[L-1], S_{A, B}[L]\}.\\
\end{array}$$  

The question: can we assume that $D(C_{A, B})$ is expected to be approximately equal to $$(1-1/e) \times 2^{1600} = 10^{481} \times 2,810560755\ldots$$ for all (or almost all) pairs of $A$ and $B$?
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.