My father is a high school STEM teacher and is pondering ideas for engineering projects where his students can model some type of physical process that demonstrates exponential growth.
One canonical example in all the HS math books models population growth exponentially but these models are idealized, only accurate for an extremely restricted time domain.
Are there natural processes that exhibit true exponential growth for all t?
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Not really a physical example, but a "real-life" one, and perhaps not exactly suited for engineering - more like uh... economics?, but do you know about pyramid schemes?
Maybe you could enjoy the opportunity to inform people about why these things don't work - as it's related to the exponential growth of the amount of people required for the scheme not to collapse.
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There is no natural process that consumes resources that is able to undergo true exponential growth for all t. A process can only consume resources that it is able to reach and a process in the universe can not reach an object that is more then $ct$ units away (where $c$ is the speed of light).
This means that long term growth is limited to $ O(c^2t^2) $ which exponential growth will eventually exceed.
Chain reaction in atomic physics: $1 \rightarrow 2 \rightarrow 4 \rightarrow 8, \ldots$
But nothing ever has or ever will exhibit exponential growth "for all t" for obvious reasons.