  We study the consequences of further modification of $f(R,R_{\mu\nu}
R^{\mu\nu}, R_{\mu\nu\sigma\rho} R^{\mu\nu\sigma\rho}) / f(R)$-theories by
means of the Dirac-Born-Infeld deformation procedure, which amounts to the
replacement of $f$ by $\lambda(\sqrt{1+2f/\lambda}-1)$ (the free parameter
$\lambda$ fixes an additional energy scale). We pay special attention to the
definition of masses of the linearized propagating degrees of freedom since
these are important to judge about the stability of the linearization around
vacuum background spaces. In this context we discuss the subtleties associated
with expanding $f(R,R_{\mu\nu} R^{\mu\nu},R_{\mu\nu\sigma\rho}
R^{\mu\nu\sigma\rho})$-Lagrangians around maximally symmetric spaces of
constant curvature, as well as with equivalence of the linearized Lagrangian to
a scalar-tensor theory. Investigation of the consequences of applying the
Dirac-Born-Infeld strategy to further modify quadratic theories, on the
stability of de Sitter vacuum, as well as its impact on the cosmological
dynamics, is the main concern of this paper.
