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Here, I am not looking for simply a repeated element, but an indefinitely repeated sequence. For example, if I have the list

1,3,4,3,4,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3

I would like Mathematica to output $3$, since the sequence $(2,3,3)$ (with length 3) repeats until the end of the sequence after a certain point. Note, the repeated sequence needs to continue until the end of the list. For example, the sequence $(3,4)$ does repeat itself near the beginning of the list, however I would not want an output of $2$ due to this repetition since the sequence $(3,4)$ does not continue after a certain point.

More formally: If I input a list for which the $i^th$ element is $x_i$, I want Mathematica to output the smallest number $n$ for which $x_i=x_i+n$ for all sufficiently large $i$.

What function/code would allow me to input a list and then would output the desired result?

Perhaps this?:

Length@Last@FindTransientRepeat[
 1, 3, 4, 3, 4, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2]
(* 3 *)

For whether the second argument 2 is sufficient for all your inputs, see FindTransientRepeat.