Palindromes in base 2 are rare but for a number to be a palindrome in some base is not so rare. For instance there are a few other bases that make 2015 into a palindrome:
Select[Range[2, 2013], # == Reverse[#] &[IntegerDigits[2015, #]] &]
{2, 38, 64, 154, 402}
For {1,1} is always b+1 palindrome, so any nontrivial palindrome base is between 2 and n-2. It is rare to not be a palindrome in any such base. The last one was 2011 and the next one is 2063.
Select[Range[1900, 2200], Function[n, Select[Range[2, n - 2], # == Reverse[#] &[IntegerDigits[n, #]] &] === {}]]
{1907, 1949, 1993, 1997, 2011, 2063, 2087, 2099, 2111, 2137, 2179}
