I'm trying to determine if there is a theoretical maximum Elo rating one player could attain in a closed group of $N$ players (assuming for simplicity that they all start with some rating $c$, such that the total Elo of the system is always $Nc$), and if there is a maximum what that maximum is in terms of $N$ and $c$ (or even if it can be lower bounded in those terms).
I believe it's easiest and potentially useful to think of the $N=2$ case first, and see whether there is a maximum difference in rating between two players which could be obtained through their playing successive games.
Elo Background
In an Elo system, each game scores such that the winner receives 1 point and the loser 0 points, with both players receiving 0.5 points for draws, and the change to a player's Elo rating is based on the difference between their actual score and their expected score, where expected score is calculated as a function of the involved players' relative Elo ratings.
More precisely, given Elo ratings $R_A$ and $R_B$, the expected scores in a match between players $A$ and $B$ are given from:
\begin{align} &E_A = \frac{1}{1+10^{(R_B-R_A)/400}} \in (0,1)\\ &E_B = \frac{1}{1+10^{(R_A-R_B)/400}} = 1- E_A \end{align}
and, given that player $A$'s actual score is $S_A$, Elo ratings are updated (for some positive constant $k$) as:
\begin{align} R_A' &= R_A + k(S_A-E_A) \\ R_B' &= R_B -k(S_A-E_A) \end{align}
where generally $k=16$.
Original Problem
Assuming player $A$ wins the game with player $B$, we can express the rating difference after $A$'s victory $d' = R_A' -R_B'$ as a function of their initial rating difference $d = R_A - R_B$:
\begin{align} d' &= R_A' -R_B' \\ &= R_A -R_B + 2k(1-E_A) \\ &= d + 2k(E_B) \\ &= d + \frac{2k}{1+10^{d/400}} \end{align}
We could then express the difference in rating between players $A$, $B$ after $A$'s winning $n$ games with a recurrence relation:
\begin{align} &d_0 = R_A - R_B \\ &d_n = d_{n-1} + \frac{2k}{1+10^{d_{n-1}/400}} \end{align}
If $\lim_{n\to \infty}d_n=\infty$ for arbitrary $d_0$ then that would of course imply the maximum Elo score in any group of players is $\infty$. This seems unlikely given that the change between terms of this sequence approaches 0 as $n\to\infty$ (the winner starts to win essentially no Elo as the number of games goes to infinity), though I'm not sure if the Elo change actually approaches 0 fast enough for the series to be convergent.
If the series does converge, however, it seems that if we could determine $\lim_{n\to \infty}d_n$ for particular $d_0$, then we could use it iteratively to calculate the maximum possible Elo in a group of $N$ players with initial ratings $c$ through first calculating how much Elo some given player $A$ is able to pull off another player $B$ (by dividing the limiting term $d_\infty$ with $d_0=c-c=0$ by 2 and adding that amount to player $A$'s rating while subtracting it from $B$'s), and then repeating this for all pairs of players $A$, $B_i$.
You could then continue to iteratively maximize among the remaining players $B_i$ into some player $B_j$, and extract from $B_j$ into $A$, but I'm not sure if this iterative process for increasing $A$'s rating would approach any particular limit in terms of $N$ and $c$. I also don't know how one would calculate $\lim_{n\to \infty}d_n$, so I'd would appreciate any help there or a nudge in a better direction if that is very nontrivial/if anyone sees an alternative way to a closed form solution for the maximum Elo rating given arbitrary $N$ and $c$.
