The problem:
We are given a group of 11 distinct positive integers, and the average of these integers is 57. We are also told that the average of the smallest 6 integers is 23. We need to find the maximum possible value of the largest integer in the group.

Approach:
To find the maximum possible value of the largest integer, we need to maximize the sum of all the integers in the group. Since the average of the smallest 6 integers is given, we can find their sum and subtract it from the sum of all the integers to get the sum of the remaining 5 integers.

We know that the average of the smallest 6 integers is 23, so their sum is 6 * 23 = 138.

To maximize the sum of the remaining 5 integers, we need to minimize the values of these integers. Since the integers are distinct positive integers, the smallest possible value for any of these integers is 1. So, the sum of the remaining 5 integers is at least 1 + 1 + 1 + 1 + 1 = 5.

Therefore, the sum of all the integers in the group is at least 138 + 5 = 143.

To find the maximum possible value of the largest integer, we need to distribute the remaining sum of 143 - 138 = 5 among the 5 integers. Since we want to maximize the largest integer, we need to minimize the sum of the other 4 integers. The sum of these 4 integers should be 5 - 1 = 4.

Calculation:
To distribute the sum of 4 among 4 integers, we can assign the value 1 to 3 of the integers and assign the value 2 to the remaining integer. This way, the sum of the 4 integers is 1 + 1 + 1 + 2 = 5.

So, the maximum possible value of the largest integer is 2.

Answer:
Therefore, the maximum possible value of the largest integer in the group is 2.