Take a hypothetical bacterium which divide once per minute. After $n$ minutes there will be $2^n$ bacteria, assuming no constraints.
But what if its growth is constrained by resources and space? I am imagining a simple model in which "food" is distributed uniformly across the plane/volume.
There is enough food in the vicinity of any bacterium to divide provided that it is not surrounded by other bacteria. In which case it will divide and the space and food around it will be consumed.
Otherwise there is enough food in the vicinity of any enclosed bacterium to keep it alive for as long as the experiment is running.
Neither food nor bacteria can move.
Assuming circular bacteria forming a honeycomb, imagine a "dent" in the border -- there will be four neighbours to the space and two gaps, therefore only two of the neighbours will be able to divide.
It's tempting to say that, although the central part is static, the perimeter will still grow exponentially and therefore the area/number will still be exponential, but I'd like an analysis which takes into account the crowding at the border. ideally for non-infinitesimal bacteria.
(I don't have a use for this, it just seems like an interesting exercise.)
If the bacteria all multiply at the same rate then you are in effect adding "area" to the bacterial colony at a rate which is proportional to the perimeter. Say the colony has a radius $R$ then the perimeter is $2\pi R$ so the rate of bacteria growth is $\lambda 2\pi R$ for some $\lambda >0$. Thus we have a differential equation
$$\frac{dA}{dt} = 2\pi \lambda R = 2\lambda \sqrt{\pi\,A}$$
According to Wolfram|Alpha the solution to this is
$$ A = \frac{1}{4}(4 B\lambda\sqrt{\pi}\, t +C^2+4\lambda^2\pi\,t^2)$$
where $B$ and $C$ are constants. So you can see that the growth rate of the number of bacteria (which is $\propto A$) is quadratic in time not exponential.