O(N^2) is not exponential but just polynomial. O(2^N) is still exponential. O(N!) and O(N^N) are faster than exponential but not that much; O(N^N) grows slower than O(N!)[1].
Hyper-exponential is most likely double exponential O(a^b^N). There is an exponential growth of apps on these platforms and another exponential growth of users on these apps.
O(N^2) is not exponential at all, it is polynomial. O(2^N) is exponential.
And the grandparent was pointing out -- correctly -- that that description doesn't really apply here. (2^N)^2 = 2^(2N) = still just 'exponential'. Plotted on a log scale it would still be a straight line.
Certainly O(N^2) is not at the top of the food chain -- for example, it is beaten by O(2^N) or O(N!) or O(N^N). All of these could, colloquially, be referred to as "hyper"exponential.
Imagine a hockey stick graph (exponential). Now imagine that hockey stick graphed on a logarithmic scale -- the logarithms more or less cancel out the exponentiation if done right, making the graph appear to be linear even though the data is exponential. Now, imagine data that, even when graphed on a logarithmic scale, still looks like a hockey stick. That is hyperexponential.
And I think I just proved Sam's point -- that it is hard to visualize/grok.