Lithium-ion batteries are currently one of the key technologies for a sustainable energy transition. However, they have a limited calendar and cycle lifetime, which are directly affected by operating conditions. Therefore, our goal is to maximize the benefits of a battery storage over its entire lifespan. Stacking multiple services (multi-use) can increase the utilization of battery storage, whereas coupling different storage technologies with complementary characteristics (hybrid energy storage systems) adds a degree of freedom for efficient and degradation-aware operation. To exploit these technological and economic advantages, we develop an energy management concept and demonstrate it in the application example of a grid-connected photovoltaic plant with hybrid battery storage. The multi-use application consists of capacity firming, participation in the electricity spot market, and peak shaving. To address the different temporal scales of the battery storage tasks, we propose a hierarchical energy management with two levels. The model predictive upper level energy management optimizes the grid power considering the time-varying electricity prices and marginal costs of battery storage operation. This multi-objective optimization problem is solved using a mixed-integer linear program with two-dimensional piecewise linearization of conversion losses and battery degradation costs. The strategy-based lower level energy management allocates power in real time to meet the grid power and ramp-rate requirements despite model and forecast errors. Extensive simulations demonstrate the advantages of the proposed approach owing to a better compliance with grid power requirements, lower conversion losses, and significantly higher benefits of the battery storage system over its lifetime.