Let the numbers in the list be $ x_1, $ $ x_2, $ $ \dots, $ $ x_n $. Then by the trivial inequality, \[(x_1 - 1)^2 + (x_2 - 1)^2 + \dots + (x_n - 1)^2 \ge 0.\]Expanding, we get \[(x_1^2 + x_2^2 + \dots + x_n^2) - 2(x_1 + x_2 + \dots + x_n) + n \ge 0.\]Since $ x_1^2 + x_2^2 + \dots + x_n^2 = x_1 + x_2 + \dots + x_n, $ \[x_1 + x_2 + \dots + x_n \le n,\]so $ \frac{x_1 + x_2 + \dots + x_n}{n} \le 1 $. Equality occurs when all the $ x_i $ are equal to 1, so the largest possible arithmetic mean is $ \boxed{1} $.