  We investigate the problem of the maximum number of cubic subwords (of the
form $www$) in a given word. We also consider square subwords (of the form
$ww$). The problem of the maximum number of squares in a word is not well
understood. Several new results related to this problem are produced in the
paper. We consider two simple problems related to the maximum number of
subwords which are squares or which are highly repetitive; then we provide a
nontrivial estimation for the number of cubes. We show that the maximum number
of squares $xx$ such that $x$ is not a primitive word (nonprimitive squares) in
a word of length $n$ is exactly $\lfloor \frac{n}{2}\rfloor - 1$, and the
maximum number of subwords of the form $x^k$, for $k\ge 3$, is exactly $n-2$.
In particular, the maximum number of cubes in a word is not greater than $n-2$
either. Using very technical properties of occurrences of cubes, we improve
this bound significantly. We show that the maximum number of cubes in a word of
length $n$ is between $(1/2)n$ and $(4/5)n$. (In particular, we improve the
lower bound from the conference version of the paper.)

Formal Languages and Automata Theory;Discrete Mathematics