  Motivated by the search for methods to establish strong minimality of certain
low order algebraic differential equations, a measure of how far a finite rank
stationary type is from being minimal is introduced and studied: The {\em
degree of nonminimality} is the minimum number of realisations of the type
required to witness a nonalgebraic forking extension. Conditional on the truth
of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of
homogeneous spaces definable in the stable theory being considered, it is shown
that the nonminimality degree is bounded by the $U$-rank plus $2$. The
Borovik-Cherlin conjecture itself is verified for algebraic and meromorphic
group actions, and a bound of $U$-rank plus $1$ is then deduced unconditionally
for differentially closed fields and compact complex manifolds. An application
is given regarding transcendence of solutions to algebraic differential
equations.
