What is the fastest elliptic curve operation f(P) in affine coordinates such that f^n(P)=P only if n is large?
With your curve, you can use the Gallant-Lambert-Vanstone (GLV) method to answer your question. Indeed, the equation of your curve is: $$y^2=x^3+7.$$ Since $p$ is congruent to $1$ modulo $3$, there are cube roots of unity modulo $p$. Let:
$$j=55594575648329892869085402983802832744385952214688224221778511981742606582254 \pmod{p}.$$ You can check that $j^3\equiv 1\pmod{p}$. The complex multiplication by $j$ sends $P=(X_P,Y_P)$ to $P'=(jX_P,Y_P)$.

Moreover, $P'=J\cdot P,$ where $$J=37718080363155996902926221483475020450927657555482586988616620542887997980018.$$

Finally, multiplication by $J-1$ can be performed efficiently (one application of complex multiplication and one addition) and has high order. Don't use $J+1$: it has order $6$.

EDIT
 $J^3$ is $1$ modulo the order of the curve, while $j^3$ is $1$ mod $p$. 

This endomorphism of the curve is the projection of the complex multiplication of the curve $y^2=x^3+7$ over the rationals to the curve reduced mod $p$. This is why is is usually called the complex multiplication.

All in all, this gives a reasonably fast way to generate random looking multiples of $P$.

The full GLV method is much more than that since it speeds up multiplication by an arbitrary constant compared to regular double and add, but its basic idea relies on having an endomorphism that can be computed quickly.