From some limited testing, it seems to me like $2^a \bmod a^3$ has no repeat values except for some powers of $2$. More generally, it also seems that $x^a \bmod (a^3+k)$ often holds to this, although a few $x$ or $k$ values do get one early repeated non-$x$-power value.
I am guessing this is basically a probabilistic result, and that $a^3$ grows fast enough that repeats become vanishingly unlikely.
Is that correct?
And if so, is it fair to label that class of problem undecidable? My impression is that there is no other way to determine whether one of these outlier $a$ values exist other than trying increasingly large $a$ indefinitely hoping for a counterexample, which seems to fit with my understanding of undecidability.