I have things to do, so I procrastinated a bit by measuring the dropout of one of my LM317 supplies, with the output trimmed to 10.60 V at 15.00 V in, and then dropping the input step by step. The regulator gave up at around 12.25 V (so a 1.65 V drop):
This was with a light load (~12 mA), and a divider consisting of 100 ohm + 2k trimmer (larger trimmers would exceed their power rating). I measured the resistances in circuit after trimming the voltage, and got 100.5 and 753.0; 100.5×10.6/(100.5+753.0) = 1.248 V so checks out.
Need to get things done now, but may do some breadboard procrastination later today 