This paper presents a novel metamaterial unit-cell configuration that may exhibit local resonance (LR) band gaps with exceptional properties — e.g., extreme width. The proposed configuration is comprised of a base wave-propagating medium to which a discrete periodic resonating branch — e.g., a branch made of a finite number of repeating diatomic unit cells — is connected. Such periodicity causes waves propagating in the branch to experience attenuation within the branch unit-cell Bragg band gap. The branch may also vibrate in resonance within its Bragg band gap due to the effect of the boundaries introduced upon truncating the nominal periodic medium. Such Bragg band-gap resonances exhibited by the branch are key to the proposed configuration as the metamaterial LR band gaps that form around them may possess exceptional properties. This paper shows that these exceptional LR band gaps are highly tunable and can be systematically designed using a semi-analytical design approach. The design approach is in part based on a recently derived analytical method that predicts, in advance, whether the branch would exhibit resonance and anti-resonance frequencies in its Bragg band-gap. Finally, a numerical case is discussed to showcase the proposed metamaterial configuration and design approach; it presents a metamaterial unit cell that demonstrates an extremely wide LR band gap. These findings open a route towards exploiting discrete, e.g., granular, periodic resonators to realize highly tunable LR band gaps.