Are there any competitors to Google AppEngine that support apps written for it?
I don't know of any hosting services, but there's an open source implementation called appscale that can apparently run on Amazon EC2. I don't have any experience using it, but they claim to be compatible with the Python API version 1.3.2, and Java API version 1.3.5.
Edit: After Matthew Crumley's excellent answer, I have substantially rewritten this answer (I didn't know about appscale before).
When you write apps to any of the cloud-specific APIs today, you're strongly locked in to the cloud vendor. Most of the business value of cloud computing comes from:
appscale may work very well, but it is far from being a proven and mature technology to build a business on. I'm not aware of a single large-scale appscale installation, and their webpage doesn't list any. So it's prudent to work from the assumption that once you build an application on a given cloud vendors proprietary storage APIs you're locked in to that vendor. The day where we can freely move a large-scale application between different cloud implementations isn't here yet.
The exceptions are the 'SQL databases in the cloud', i.e. Amazon RDS (hosted MySQL) and Windows Azure's SQL API (hosted SQL Server). If you build a traditional 2-tier web application on top of these, then you can move it to traditional hosting with a MySQL / MSSQL backend. But that's outside OPs question, since Google AppEngine doesn't offer a MySQL backend.