Catenary curves are the sum of two exponentials -- One concave upward but decreasing, and the other concave upward and increasing. They are a good model for the curve of suspension bridge cables. (The key assumptions are that the weight of the cables is small compared to the weight of the bridge deck, and the weight of the bridge deck is constant along the length of the bridge.)
Amplitudes of bouncing motions (and pendulums) tend to have exponential decay.
The shape of absurdly tall buildings or space elevators. If you tried to build a building many miles high, how would the area of the building have to change as you went up, just to support its own weight? If you haven't gone up far enough for the acceleration of gravity to change much, the curve will be an exponential. If you try to make an elevator cable extending down from geosynchronous orbit, the cross-section would also vary exponentially while the cable was still "short". But if you try to reach down to the planetary surface, the change in the acceleration of gravity (adjusted for centrifugal acceleration) would distort the exponential into something like a Gaussian curve. A longer overview (and a short bibliography) are in Hans Moravec's article "Cable Cars in the Sky", in The Endless Frontier.