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.
