  We generalize the recently introduced extended scalar-tensor-Gauss-Bonnet
(ESTGB) theories and their close relatives to include spontaneous growth of
nonscalar fields such as vectors. This is analogous to the program that
developed spontaneous tensorization from the original spontaneous scalarization
theory of Damour and Esposito-Far\`ese (DEF). The new larger family of theories
conserves the appeal of the DEF theory in terms of conforming to weak-field
tests and also providing large signals in strong gravity. Moreover, they
provide a much richer phenomenology including spontaneous tensorization of
black holes as in ESTGB. These theories, together with other possible future
extensions that we discuss, testify to the ubiquity of spontaneous growth in
gravity. We also note that theories with derivative coupling require special
attention since they can lead to potentially problematic higher derivatives in
the equations of motion.
