What is the maximum number of different three letter palindromes you can make using any letter of the alphabet?

The maximum number of different three letter palindromes that can be made using any letter of the alphabet is 1 - that letter repeated 3 times.

However, if you meant you can pick any letter for each letter of the palindrome, then:

The first and last letters are the same and there is a choice of 26 letters for them;

For each choice for the first (and last) letter, there is a choice of 26 letters for the second letter

Making a total of 26 × 26 = 676 such three letter palindromes (where case of the letter does not matter).