High performance computing architectures combine massive parallelism with heterogeneous hardware; numerical modeling codes running on such platforms are increasingly constrained by memory and communication bandwidth. While discontinuous Galerkin (DG) methods are attractive in this setting because of their locality and high arithmetic intensity, their practical performance is often limited by the cost of moving and operating on long solution vectors stored at full precision. Many mixed-precision techniques attempt to address this issue using reduced precision or generic compression, but such approaches, while helpful in some use cases, do not fully exploit the spectral accuracy decay naturally inherent in modal DG solution representations. In this work, we introduce a family of degree-aware adaptive spectral block floating-point (ASBFP) formats for modal DG discretizations. These formats exploit the spectral decay of polynomial approximation in DG methods when assigning numerical precision: low-order modes are stored with high fidelity, while higher-order modes are represented with progressively fewer mantissa bits according to a discrete, degree-dependent allocation rule combined with a shared or hierarchically biased exponent. In the present work, ASBFP is implemented as a drop-in replacement for the DG solution vector internally employing a compressed storage format with a decode-operate-encode workflow which relies on the standard floating-point arithmetic on CPU, GPU, and FPGA platforms for numerical computations. We demonstrate the method on one- and multi-dimensional DG discretizations of representative hyperbolic problems in computational fluid dynamics, covering smooth and non-smooth transport problems as well as Euler equations. Numerical experiments show that ASBFP substantially reduces memory footprint and memory traffic while maintaining accuracy comparable to standard IEEE floating-point formats. These results highlight the potential of adaptive, degree-aware reduced-precision floating point formats for bandwidth-limited exascale systems.