You can make it somewhat objective, e.g. a "Martin-Löf random" real number is a number where no algorithm can predict the next digit from previous digits better

You can make it somewhat objective, e.g. a "Martin-Löf random" real number is a number where no algorithm can predict the next digit from previous digits better than chance. A specific real number can have that property, in fact most of them do. (Before you ask, pi is not random in that sense, because there's an algorithm that predicts digits of pi with 100% accuracy.)