  A word is called closed if it has a prefix which is also its suffix and there
is no internal occurrences of this prefix in the word. In this paper we study
words that are rich in closed factors, i.e., which contain the maximal possible
number of distinct closed factors. As the main result, we show that for finite
words the asymptotics of the maximal number of distinct closed factors in a
word of length $n$ is $\frac{n^2}{6}$. For infinite words, we show there exist
words such that each their factor of length $n$ contains a quadratic number of
distinct closed factors, with uniformly bounded constant; we call such words
infinite closed-rich. We provide several necessary and some sufficient
conditions for a word to be infinite closed rich. For example, we show that all
linearly recurrent words are closed-rich. We provide a characterization of rich
words among Sturmian words. Certain examples we provide involve
non-constructive methods.
