For the sake of adding to my own discussion, my issues (specific to AWS) were:
1. Installing dependencies became complicated. When Chalice worked, it was great. Certain dependencies (in my case lxml) required a precompiled bundle that someone had thankfully already taken care of packaging.
2. Visibility into the process was poor. It felt hard to get into the system and really introspect it once it was deployed.
3. I felt pushed to other "serverless" solutions within AWS that I wasn't comfortable with. I understand the inherent vendor lock-in with the current serverless offerings, but I felt like I was doubly locked in when e.g. the "best" choice for a db was Amazon Dynamo.
4. Current serverless frameworks feel very minimal. I'm a big fan of both Django (batteries included) and Express (bring everything on your own). The tech that is out there now feels like Express, except that there isn't a lot of extensions to layer on top. You sort of have to cobble your own together still.
5. Lack of Best Practices. I just didn't see a ton of thought leadership on how to do things the right way in a serverless environment. I might have just missed this though.
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Overall, I've come away impressed but I feel like building something entirely on serverless isn't quite as fleshed out as it needs to be yet. I think it's very close though.