Rich words (also known as full words ) are a special family of finite and infinite words characterised by containing the maximal number of distinct palindromes. We prove that the number of rich words of length n over a finite alphabet A (consisting of 3 or more letters) grows at least polynomially with the size of A. We also show asymptotic exponential growth for the number of rich words of length 2n over a 2-letter alphabet. Moreover, we discuss possible factor complexity functions of rich words and consider the difficult (open) problem of enumerating the finite rich words over a fixed finite alphabet
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