2001 Ford F-250 Super Duty with 7.3L Diesel
Battery light is on constantly after one of the positive battery cable ends got broke.

Things done since cable broke:

1. Replaced positive cable to both batteries and to starter (all one piece) $100
2. Replaced both batteries.
3. Replaced serpentine belt.
4. Replaced broken belt tensioner.
5. Had alternator tested (tested good) on vehicle.

What else would cause this light to stay on all the time?Attachment 48646

The only thing that can turn that light on is either a faulty alternator or the LG/RD wire has a short to Ground.
When you say the alternator was tested, do you know if that particular internal circuit was verified?

The problem is likely with the alternator. In most cases, replacing the external voltage regulator/brush assembly will solve the problem. It's an easy job--disconnect the negative battery cable and use a Torx 20 bit to remove the screws mounting the voltage regulator brush assembly. Using a paper clip or drill bit, to temporarily hold the compressed brushes in place, simply install the new assembly and button things up. My bet is the brushes are worn down too far.

While you are at it, clean both ends of the alternator cable, the battery ground going to the frame, and the 150 amp mega fuse connections on the end of the fuse box. One final thing: check the mega fuse with a multi-meter. While the engine is running, the regulated battery voltage across the battery terminals should be 14 to 15 volts.

I agree Brushes will do that, and lite may go on and off before total failure.