Base64

Video / audio codecs.

Cryptographic algorithms. (encrypt / decrypt, hash functions)

Cryptographic random number generators.

Ordinary random number generators.

PKI (key generation, validation)

Packing / unpacking binary structures used in wire protocols or message formats. (Example: a certificate, public key, jpeg, protocol buffers)

Implementing things like Protocol Buffer. (Language neutral binary format)

Implementing BigInteger. JDBC drivers.

Java libraries that create, compile, decompile or otherwise manipulate Java Byte Code.

Java libraries that create or manipulate other types of binary executable code. (eg, an EXE file)

Implementing Class Loaders for java.

The reasons you might work with bytes in Java are as endless as why you would work with bytes in C or any other language.