How can $ww = www$ hold for any word $w$? Speaking in terms of automata and regular languages, how would it be possible for a string repeating some $w$ twice equal a string repeating that same $w$ thrice? That is, why is the language

$\qquad L = \{w \in \Sigma^ * \mid ww = www\}$

not empty? The only thing I can think of is $w = abab, ww = abababab, www = abababababab$, but I don't think this is correct.
The only way that $w w = w w w$ is that $w = \epsilon$. Algebra of strings (for mathematician types, the free monoid on $\Sigma$) isn't that different from multiplication...