When something grows exponentially, its value at each step is the same constant factor times its value at the previous step. 1, 2, 4, 8, 16 is an example in which the constant factor (the “power”) is 2. To say that a quantity “grows exponentially” is just to say that its growth over time has this particular quality: a constant ratio of change. Where the power involved is greater than 1, not only does the quantity itself grow without limit, the rate of growth grows without limit, as does the rate of growth of the rate of growth, and so on. If the power is significantly greater than 1, the quantity becomes enormous remarkably quickly.
This association of “exponential” with explosive growth leads people to say—incorrectly and annoyingly—that any sudden or dramatic growth is “exponential.” The usage is bothersome in multiple ways. First, not all exponential growth is rapid at first. Where the exponent is just barely greater than 1—let us say, for example, 1.0000001—the quantity involved will grow quite slowly for quite some time. Second, where you have only two data points (small before and large after), almost any function works just as well as an exponential function. You could call it “logarithmic growth” or “quadratic growth” or “sinusoidal growth” and be just as accurate. Calling it any of these names misses the point that you don’t have enough information to determine the mathematical quality of the growth.
If you have a series of values that are approximated by an exponential curve, then by all means, go ahead and refer to the “exponential growth.” (And if you have a curve that grows exponentially backwards in time, call it “exponential decay.”) But if all you have is a sharp increase, find a name for it that doesn’t already have a precise mathematical meaning. “Sudden and dramatic growth” is perfectly fine, as is referring to an “x percent increase” for some impressively large value of x But please do not perpetuate the bad habit of referring to every increase as “exponential.” Not all crunches are sickening; not all growth is exponential.