I think I can give a characterization of your limit as a sum of Teichmüller representatives.

Let q = p^f be some power of p. Let Z_q = W(F_q) be the valuation ring of the unramified extension of Q_p of degree f. Then for any a in Z_q, there is a unique root of x^q - x in Zq congruent to a mod p. One can identify this with the limit, as n tends to infinity, of a^{q^n}.

I've never seen this before, but I guess you can do the same thing even if your extension is ramified. Let R be some finite extension of Z_p. Let F_q denote its residue field. Then for any a in R, there is a unique root of x^q - x in R congruent to a mod p^(1/e), where e is the ramification index. Again, it can be idenified with the limit of a^{q^n}.

Assuming the limit you mentioned exists, it is the same as the limit of a_1^{q^n} + ... + a_k^{q^n}. And then this limit is the sum of the Teichmüller representatives that I just described.

I think I can give a characterization of your limit as a sum of Teichmüller representatives.

Let $q = p^f$ be some power of $p$. Let $Z_q = W(F_q)$ be the valuation ring of the unramified extension of $Q_p$ of degree $f$. Then for any $a$ in $Z_q$, there is a unique root of $x^q - x$ in $Z_q$ congruent to $a$ mod $p$. One can identify this with the limit, as n tends to infinity, of $a^{q^n}$.

I've never seen this before, but I guess you can do the same thing even if your extension is ramified. Let $R$ be some finite extension of $Z_p$. Let $F_q$ denote its residue field. Then for any $a$ in $R$, there is a unique root of $x^q - x$ in $R$ congruent to $a$ mod $p^{1/e}$, where $e$ is the ramification index. Again, it can be identified with the limit of $a^{q^n}$.

Assuming the limit you mentioned exists, it is the same as the limit of $a_1^{q^n} + \cdots + a_k^{q^n}$. And then this limit is the sum of the Teichmüller representatives that I just described.