Start with three numbers $1,1,1$, each operation replaces one of the numbers with the sum of the other two. After 10 operations, what is the maximum possible value of the largest number among the three obtained?
8. Answer: 144

Analysis: Each time, sort three numbers in ascending order, then replace the smallest number with the sum of the other two numbers, the result is $\{1,1,1\} \rightarrow\{1,1,2\} \rightarrow\{1,2,3\} \rightarrow\{2,3,5\} \rightarrow\{3$, $5,8\} \rightarrow\{5,8,13\} \rightarrow \cdots$

Upon observation, the largest number forms a Fibonacci sequence, starting with the numbers 1, 2, and from the third term onwards, each number is the sum of the two preceding numbers. Therefore, it is
$$
1,2,3,5,8,13,21,34,55,89,144
$$

After 10 operations, the 11th number in this sequence is 144, which is the maximum possible value of the largest number.