I've jumped with one stirrup tons of times! ...just not necessarily on purpose.
In all seriousness, I do remember a lot of lessons with no stirrups, bareback, airplane arms, etc. Unfortunately I wouldn't trust any of the horses I ride with that right now. I tried airplane arms while bareback on the flat with my favorite horse not too long ago... I trust him so much, but even he went, "Oh cool! No reins! Let's go!"