Hallo Rob,

I think this high output voltage is typical for this 78XX series, when having two conditions fullfilled:

1. Much too high input voltage resulting in voltage drop accross regulator of much more than 3V.

2. No load at output.

I have seen this many times in the past. Because of this I always connect a resistor of less than 10kOhm at output, when 'no load' condition is possible.
Interesting: I found many times 7915 failing, but rarely 7815.

I just had a look at datasheet and found to my surprise, that all specifications about output voltage refer to load currents bigger than 5mA!

Regards,
Kai