I don't know anything about a 13 button accordion owned by Steve Riley, but link #1 will lead you to a demonstration of the Canadian "Accordéon Bouchard", which has 14 buttons.
Note all the different keys you can play on that instrument.
Link#2 shows Steve with Doug Kershaw playing Lake Arthur Stomp.
I can't see or hear if he's using the three plus-buttons.
I heard the first time at the festival Saulieu 2013 that Eric was building an accordion like this.
It's discussed earlier here, so if you want to know more , search for it with Eric Martin.
Bryan's description of the layout of Steve's is correct. The one Bryan built for me is different, more chromatic. It also has 5 stops and another set of bass reeds to play in 3rd position. It's #29 on Bryan's L'Anse Grise Accordion web site. I've got a video of Chris Miller playing mine at Augusta, but no idea how to post it here.
These are accordeons made by and for professionals with mitered corners and exquisite workmanship. Less than the price of the average top 3 Louisiana made top three accordeons.