The amount of money you have to grow your business is directly proportional to the cost overheads of running your app minus the revenue it generates. We are assuming that the larger your internet service is the more money you will make.
A service that processes tweets from the firehose in realtime. Assuming you decide to process everything day 1, thats a case where the overhead from running your app could greatly outweigh the revenue it generates.
I mean, I guess? I would question that business and processing model heavily, though. Most sustainable business models show growing from some kind of smaller MVP to a larger full-featured application.
Significant overhead costs to go from nothing to steady-state instantaneously would be enough to scare me.
It's an interesting question as it was exactly a concern we had when we started to design the product we're currently working on (visualops.io). We realised that many people struggle hiring one (or more) ops people when they feel they're not big enough, although already too big to actually keep going without.
Usually, at that point, from our experience, we see two types of choices: 1- as you're suggesting, hire an actual ops person (but switching from a platform like Heroku to something like AWS where everything should be configured manually, and constantly updated, may require more than one person in some cases...) 2- keep on using platforms like Heroku (and get ripped off).
We have been ourselves in this situation when we have launched the first version of our product, and are now happy users of our own solution ;)
I don't think with the way technology is now, even without utilizing all the advanced services that AWS offers (so simply using the basics like ec2 / s3 / ebs / etc) you need a full full time ops person until you're paying 50k/month + in server hosting, on AWS reserved instances.
Well it all depends of your application and which level of automation you're looking for. If your growth require you to often scale your infrastructure, and you want to save your developers to deal with operations, then you may need someone.