  Recently, a new nonlinear mechanism for black hole scalarization, different
from the standard spontaneous scalarization, was demonstrated to exist for
scalar Gauss-Bonnet theories in which no tachyonic instabilities can occur.
Thus Schwarzschild black hole is linearly stable but instead nonlinear
instability can kick-in.
  In the present paper we extend on this idea in the case of multi-scalar
Gauss-Bonnet gravity with exponential coupling functions of third and fourth
leading order in the scalar field. The main motivation comes from the fact that
these theories admit hairy compact objects with zero scalar charge, thus zero
scalar-dipole radiation, that automatically evades the binary pulsar
constraints on the theory parameters. We demonstrate numerically the existence
of scalarized black holes for both coupling functions and for all possible
maximally symmetric scalar field target spaces. The thermodynamics and the
stability of the obtained solution branches is also discussed.
