No it doesn't. It means an exponent is present.
Constant growth rate gives rise to a quantity that is an exponential function of time [not to be confused with a constant rate of increase, which gives rise to a linear function of time].
As that Wikipedia article says, the continuous-time equation for exponential growth, x(t) = x(0) e^(kt), arises as the solution to the ODE x'(t) = kx, where k is the constant growth rate.
Similarly, in discrete-time, exponential growth follows the equation x_t = x_0 (1+r)^t, where r is the constant growth rate.
I stand corrected.
The formula for constant growth over time, for example, a 2% increase every year, would be starting salary * 1.02 ^ t where t is the number of years. So there is an exponent present, and constant growth is an exponential process.
;).