  We demonstrate the existence of a family of finitely generated subgroups of
Richard Thompson's group $F$ which is strictly well-ordered by the
embeddability relation in type $\epsilon_0 +1$. All except the maximum element
of this family (which is $F$ itself) are elementary amenable groups. In fact we
also obtain, for each $\alpha < \epsilon_0$, a finitely generated elementary
amenable subgroup of $F$ whose EA-class is $\alpha + 2$. These groups all have
simple, explicit descriptions and can be viewed as a natural continuation of
the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr
\mathbf{Z}$, and the Brin-Navas group $B$. We also give an example of a pair of
finitely generated elementary amenable subgroups of $F$ with the property that
neither is embeddable into the other.
