  We study two--generated subgroups $\langle f,g\rangle<\mathrm{Homeo}^+(I)$
such that $\langle f^2,g^2\rangle$ is isomorphic to Thompson's group $F$, and
such that the supports of $f$ and $g$ form a chain of two intervals. We show
that this class contains uncountably many isomorphism types. These include
examples with nonabelian free subgroups, examples which do not admit faithful
actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit
faithful actions by $PL$ homeomorphisms on an interval, and examples which are
not finitely presented. We thus answer questions due to M. Brin. We also show
that many relatively uncomplicated groups of homeomorphisms can have very
complicated square roots, thus establishing the behavior of square roots of $F$
as part of a general phenomenon among subgroups of $\mathrm{Homeo}^+(I)$.
