  A systematic search for extremely deformed structures in the N~Z nuclei of
the A~40 mass region has been performed for the first time in the framework of
covariant density functional theory. At spin zero such structures are located
at high excitation energies which prevents their experimental observation. The
rotation acts as a tool to bring these exotic shapes to the yrast line or its
vicinity so that their observation could become possible with future generation
of $\gamma-$tracking (or similar) detectors such as GRETA and AGATA. The major
physical observables of such structures (such as transition quadrupole moments
as well as kinematic and dynamic moments of inertia), the underlying
single-particle structure and the spins at which they become yrast or near
yrast are defined. The search for the fingerprints of clusterization and
molecular structures is performed and the configurations with such features are
discussed. The best candidates for observation of extremely deformed structures
are identified. For several nuclei in this study (such as $^{36}$Ar), the
addition of several spin units above currently measured maximum spin of
$16\hbar$ will inevitably trigger the transition to hyper- and megadeformed
nuclear shapes.
