For what natural $n$ does there exist a square composed of $n$ squares?

Note first that you can always add three squares to a configuration by splitting one sub-square into four.

Then take a $3\times 3$ square out of the corner of a square of side $4$ to find a configuration with $8$ squares.

This gives $1+3n; 6+3n; 8+3n$ as possibles, leaving just $2,3,5$ as impossible.

See also the comment on cute squares in this link: https://en.wikipedia.org/wiki/Squaring_the_square which says it can be done with squares of no more than two sizes for all positive integers other than 2,3,5.