The usual S-curve would only look S-shaped in a linear chart (like BTCUSD does). In a log-chart the S-curve looks mostly linear, as its steep-rise phase is determined by exponential growth.
The s-curve or logistic function is just tanh by another name (merely scaled and shifted).
Anyway, take the log of the logistic function and this is what you get:
http://m.wolframalpha.com/input/?i=y%3Dlog%281%2F%281%2Be%5E%28-x%29%29%29%2C-10%3C%3Dx%3C%3D10%2C-10%3C%3Dy%3C%3D1&x=11&y=7Right now bitcoin is in the part of the curve on the left, where you see the sloping blue line.
Exponential growth won't last forever. It can't, because the number of potential users and the amount of wealth in the world is finite.