Consider a sequence $a_k(n)$, where $a_k(1) = 1$, and $a_k(n) = a_k(a_k(n-1) \pmod {n-1}) + k$ for integer $k > 1$.
For the first few $k$, the sequence quickly stabilizes, ending in an infinite repeating string of numbers, almost always of period $1$.
I'm I haven't proved whether the sequence starts infinitely repeating for all $k$, though, and I don't know if the length of time it takes to stabilize grows quickly with $k$, or how quickly it grows.
The first few sequences are as follows:
$a_2(n) = 1,3,3,5,3,5,5,5,5,5...$ Stabilizing by $6$ steps.
$a_3(n) = 1,4,7,4,7,7,4,7,7,7,7,7...$ Stabilizing by $8$ steps.
$a_4(n) = 1,5,5,9,5,9,9,9,5,9,9,9,9,9...$ Stabilizing by $10$ steps.
$a_5(n) = 1,6,11,11,16,6,11,16,21,16,11,16,16,16,11,16,21,16,21,11,16,21,21,21,...$ Stabilizing by $22$ steps.
A second related sequence, $b_k(n)$, is defined with iterated multiplication by k instead of addition. $b_k(1)=1$, and $b_k(n) = k*b_k(b_k(n-1) \pmod {n-1})$.
The sequences for $ k=1, 2, 3$ stabilize quickly into a repeating pattern, but I have no idea whether or not the sequence stabilizes for any other $k$.
$b_2(n) = 1,2,4,2,4,4,4,4,4,4...$ Stabilizing by $5$ steps.
b_3(n) = $1,3,3,9,3,9,9,9,3,9,9,9,9,9...$ Stabilizing by $10$ steps.
Does this sequence stabilize for $k=4$ or greater?