Hmm, let's see... I bent the rim in Death Valley, about 14-15 miles from pavement and the front K60 came off the bead. Some rocks and the compressor got the bead back in place and the tire aired up enough to get back to the hotel, then a little hammer action and I rode it to Vegas to have it properly straitened.
I've probably fixed half a dozen flats by the side of the road with nothing more than the compressor and sticky string kit.
I took off on a weekend ride with some friends and thought I had about 2k left on the PR3 Trail, (what a pos tire!). I was wrong, the tire was at the wear bars in 250 miles. Bare in 50 more and cords in another 80 or less. That led to an unexpected rear tire purchase on the road. It was cheaper, and a lot faster, for me to pull the wheel off myself in the parking lot than to pay the dealer to do it and wait for a spot in line. That did require the wrenches to come out.
I prefer to be prepared to self recover, wether it be due to shit just happening, or me misjudging or just doing something stupid... like thinking that 'good line' was ok to ride a 45 mph.
Under some conditions you can get away with riding a flat tire for a ways. But they are destroyed too, and can come off the bead and lock the wheel. It gets ugly about there.