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 $f$.

What about $\sigma = \pi^2$, or $\sigma = \pi^k$ for $k > 2$? 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.

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 $k > 2$? 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.