Patrick,
Do you really mean the penalty should increase exponentially? Do you mean the penalty, as in penalty strokes, should be exponentially higher? Like 10 times higher? Or are you saying that the resultant shot should be harder, or twice as hard, or 10 times as hard following a miss on a short hole?
The obvious way to achieve an exponential penalty would be to have penalty strokes closer at hand on short holes - water, OOB, unplayable lies, lost ball jungles. Seems to me this would make such holes not much fun. Why ever risk the exponentially higher score.
If you mean that a short hole should present a birdie opportunity, where a miss would likely result in making a bogey, then, yes, that seems like a good trade off to me. But it doesn't seem to require an exponentially more difficult second or recovery shot. Just a harder shot to get up and down. If you want to guarantee a double or worse, then you'd likely need to introduce a penalty shot.
One short hole that comes to mind and fits the bill is the Postage Stamp at Troon. Hit the green and the reward is a potential birdie. Miss the green and you can run up a score going back and forth between bunkers (perhaps running up even an exponential score
).