Plenty of binary formats out there containing big endian integers. An example is the format I'm dealing with right now: ELF files. Their endianness matches that

Plenty of binary formats out there containing big endian integers. An example is the format I'm dealing with right now: ELF files. Their endianness matches that of the ELF's target architecture. Apparently they designed it that way in order to ensure all values in the ELF are naturally encoded in memory when it is processed on the architecture it is intended to run on. So if you're writing a program loader or something you can just directly read the integers out of the ELF's data structures and call it a day. Processing arbitrary ELF inputs requires adapting to their endianness though.