I have spent the last year and a half building a completely serverless production service on Lambda, API Gateway, and DynamoDB (along with the standard auxiliary services like CW, SNS, Route53, S3, CF, X-Ray, etc.). It was a lot of work establishing new patterns for many of the operational aspects, particularly custom CW metrics and A/B deployments with Lambda traffic shifting, but in the end everything is set up nic…
This does come with additional cost. Serverless pricing doesn't scale, your costs increase linearly with your usage, and there're no discounts for bulk usage or reserved pricing.
We were recently on the receiving end of a massive HTTP GET Flood DDoS and although we did not experience any downtime as a result of it, I ended up finding out about it a few days later when billing alarms started going off.