'Growing exponentially' is an overused phrase, and people actually forget what it means. Exponential growth makes a lot more sense when you're describing the number your user base, where generally speaking each user is worth the same to you. How sustainable is actual exponential monetary growth? I mean, I can't speak for Stripe, but I seriously don't think they're getting exponential growth in transaction.
That said, most naturally occurring growth is exponential. In the case of a webservice, whenever a new user increases the chance of another new user signing up by at least some constant factor, no matter how small, you have exponential growth. Things working against that are churn and later, like in the case of Facebook, the saturation in your market. If you discount for those factors (I don't think they hugely affect Stripe right now anyway), EVERY application that has even the tiniest viral co-efficient, even just traditional word of mouth, will grow exponentially.
The specific case of Stripe transactions is particularly interesting. They grow in both the number of companies using Stripe as well as the size of each of those companies. I have no knowledge of Stripe metrics, but I am fairly certain they've seen exponential growth over the past months. Much like Square btw ;)
