There are none between 2 and 9 (inclusive) unless a 1-digit number is a palindrome of itself, in which case there are 8.
Between 11 and 99 (discounting repeated digits) we have 9 palindromes 11, 22, ..., 99.
With 3 digits we have aba as the pattern. For a=1 and b=1-9 we have 9 palindromes. Similarly for a=2, b=1-9; a=3, b=1-9; a=9, b=1-9. So we have 9×9=81 3-digit palindromes.
In total there are 9+81=90 palindromes between 2 and 999 (inclusive). If single digits are considered to be palindromes, there are 90+8=98 palindromes.
If same-digit numbers are disallowed (that is, a, bb, ccc types, 26 of these) then there are 72 palindromes.