Somewhat related to this question, but I'm interested in a phrase, not just a single word.
The longest I am aware of is
A man, a plan, a canal: Panama
Are there longer palindromic phrases?
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There is no upper limit to how long a palindrome can be. Poems have been written that are palindromes.
In English, two palindromic novels have been published: Dr Awkward & Olson in Oslo by Lawrence Levine (1986, 31954 words), and Satire: Veritas by David Stephens (1980, 58795 words). In French, Oulipo writer George Perec's "Grand Palindrome" (1969) is 5,556 letters in length. In Hebrew, Ghil'ad Zuckermann wrote a 153-word palindromic story called "Lear's in Israel."
For much shorter palindromes: The first place I look for constrained writing is Mike Keith's website, and it has a few short palindromes and a story about them, though neither is as impressive as some of his other work.
Peter Norvig discusses this here, suggests his own, and links to another possibility.