  We study infinite binary words that contain few distinct palindromes. In
particular, we classify such words according to their critical exponents. This
extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with
18 and 20 palindromes happen to be morphic images of the fixed point of the
morphism $\texttt{0}\mapsto\texttt{01}$, $\texttt{1}\mapsto\texttt{21}$,
$\texttt{2}\mapsto\texttt{0}$.
