The example that Brian always gave was a "hex" data type, so that you can store MD5 numbers using 128 bits of storage instead of 256. A human readable MD5 is a 256 bit string (32 characters) representing 128 bits of actual data. So storing MD5 values in the database (a very common thing) requires twice as much space as it should.
Now, the question I've always asked is, "Why did nobody actually implement the type?". It seems really useful. And it was always said to be trivial. So why doesn't it exist?
Perhaps because you can use a BINARY(16) column and store the raw value, converting between hex and binary? Or maybe just use a UDF for storage and retrieval?
Should a hex data type get added after all this time? How about a UUID data type? Especially with UUID being common, it could save a lot of space.