I drove like 80km once on a dead alternator, with the car being turned on and off several times in this period. My car uses a pretty big battery, but i think it also uses extra power on startup etc.
I was pretty sure my alternator was dead but my mother told me to keep driving it till it dies incase the battery just got flat somehow (bad advice). Eventually the steering and the brakes and the traction control started acting up a little bit (not too bad, but enough to make driving around other people uncomfortable).
Unsurprisingly, after a while it just wouldn't turn on, so I hill started it. I also thought that once a car is started it doesn't need battery power, but it seems that's wrong because eventually (.. on a main road) the engine just lost all power and it stalled and wouldn't start up again even with a hill start.
The tow truck did the rest
My old car (no fancy stuff, carbie etc) could drive without a battery i think, because we used to "hot swap" the battery (ie change batteries with the engine running) with no trouble. I guess new cars need the battery for the ECU.
So if you just need to do 1 mile you should easily be fine. However, incase the brakes and traction control go mad, drive carefully and try do it when no one else is on the road.
If you wanna be 100% extra safe u can take the battery out of another car and keep it in the passenger seat, incase your one dies. Make sure you keep the battery upright