Let's define the problem step by step to solve for the greatest possible percentage of questions Amy could have answered correctly.

### Solution By Steps

**Step 1: Analyze Amy's performance**

- Amy answered the first three questions wrong, but she answered all the other questions correctly.

- Therefore, the first three questions are incorrect for Amy, and all subsequent questions (starting from question 4) are correct.

**Step 2: Analyze Scott's performance**

- Scott answered exactly two questions wrong: the last question and the question before the last correct answer.

- This means Scott answered the second-to-last question wrong and also the last question wrong.

- All other 26 questions before these two wrong answers were correct.

**Step 3: Determine the overlap in wrong answers**

- We are told that Amy and Scott were both incorrect on exactly one common question.

- From Amy’s incorrect answers (questions 1, 2, and 3) and Scott’s incorrect answers (the last two questions), the only possible overlap occurs if Amy’s third wrong answer matches Scott's second wrong answer (i.e., the second-to-last question of the test).

- Thus, Amy's third wrong answer is the second-to-last question, which means she got question number \( n-1 \) wrong.

**Step 4: Calculate the total number of questions**

- Scott answered the last 26 questions before his last wrong answer correctly, and he got the last two questions wrong.

- Therefore, the total number of questions on the test is \( 26 + 2 = 28 \).

**Step 5: Count Amy's correct answers**

- Amy answered the first three questions wrong (questions 1, 2, and 3).

- Additionally, she also answered the second-to-last question wrong (question 27).

- Hence, Amy answered \( 28 - 4 = 24 \) questions correctly.

**Step 6: Calculate the greatest possible percentage of correct answers**

- Amy answered 24 questions correctly out of 28 total questions.

- The percentage of correct answers is calculated as:

\[

\text{Percentage} = \left( \frac{24}{28} \right) \times 100 = 85.71\%

\]

### Final Answer

The greatest possible percentage of the total number of questions that Amy could have answered correctly is **85.71%**.