If not tipped, how tall will triple crown blackberry canes get the first year?
I top mine at about 8 feet, and I have to do that multiple times a season to keep them there. If properly supported, I'd suspect that they'd probably get to twice that height... But obviously you wouldn't want that. I believe topping helps stimulate lateral growth on which the fruit grows.
Extremely straight 3c canes will go up ten feet before beginning to arch over, so Glenn's estimate is good.
These heights are for an established planting, 3 years or more. Unlikely to happen for new plantings.