Two tubes, patch kit and frame pump. always. In my sewup days, good spare, patch kit and frame pump. (Zephal HPs then, HPXs now.)
I've never owned or used either a mini pump or CO2. I have had rides where I've used most of my resources. (Both tubes and several patches one ride. Granted, that was because both my brand new latex tubes blew out at the valve/tube juncture; something I have never seen before or since. (And no, not from the valve being manhandled with a pump. The tube was glues to a short piece of heavier tube with the valve. That glue joint failed in a slow blowout. Both tubes 10 miles into a 100 mile ride and a mile apart. Got another (or two, I forget) routine flat. Stopped 15 miles from home at a Fred Meyer (the store that carries everything) and bought a miserable tube and patchkit - just because. It worked, No more flats.
A thousand years ago, I rode from Boston to Fitchberg to watch the Longsjo Classic and the country's best racers. Got my second flat on the way home and patched the sewup on the side of the road 30 miles from home.
Now those are my most notable success stories re: flats - over 200,000 miles of riding. I look at it as good ol' Murphy's Law. That being prepared goes a long ways to stuff NOT happening. But even so, sooner or later, it does.
Oh, and frame pumps! What a blessing! They allow you to be tired, spacey, clumsy and on and on. Screw up and pinch the tube?. Have a really bad day with thorns? Patch it and try again, You can do this until you run out of patches. (Did you bring that much CO2? Are you willing to pump up that many times with a mini pump? And being able to stop and top off pressure just because. Get to gravel? drop the pressure, Back on pavement, Pump it back up. Racing a slow leak home? Want 110 psi, not 105? No problem. (In my racing days, I lent my frame pump to a skinny engineer/ post 40 yo time trial rider to pump his race tires to 120 psi. He had no trouble doing it. That pump and the others on my other bikes were the only pumps I owned when I raced. No floor pump, no compressor, no CO2. All my tires were sewups. Race tires had latex tubes and needed to be pumped from near scratch every time.
The manufacturers of carbon fiber bikes have done their best to obsolete frame pumps but as long as pneumatic tires of most of 100 psi are critical, good frame pumps will always be a blessing for those wiling to carry them. (And with all the freedom CF gives in design, why doesn't someone make a CF frame with sexy, near integral TT frame pump stowage? They've done everything else - except get around this issue of needing high pressure air.)
Ben (off on one of his favorite rants)