Johnny MNEMONIC (1995)

Johnny is a data trafficker who has an implant that allows him to securely store data too sensitive for regular computer networks. His brain can carry nearly 80 gigabytes worth of data, or 160 gigabytes if he uses a doubler. Johnny uses this implant to act as a courier between contracting parties. On one delivery run, he accepts a package that not only exceeds the implant's safety limits (and will thus kill him if the data isn't removed in time), but also proves to contain information far more important and valuable than he had ever imagined. He has to remove the data and avoid being killed by assassins sent after him by the company who owns the data.