I tried the series with different limit (plotting the series upto a value). I found it interesting to see that the pattern remains similar at all limits - A steep hockey stick curve. Like a simple fractal pattern. Many successful social/web user growth has a similar growth curve. YouTube had it in early days, FB did and I guess now Google+ does.
Here is the visualization.
Tools used: 5 lines of python code to generate the Fibonacci series. MS Excel to generate the charts. I generated multiple charts and manually placed them together.