I am currently making a video game in Java that will most likely be distributed through Steam. Of course, many Steam users won't have Java already installed in this case. If this happens, I can either have a small native launcher that shows an informative dialogue telling the user they need to install Java, or simply bundle my own Java runtime environment (specifically OpenJDK 8) with my game. My game will be closed source, however it is based on an open source (BSD licensed) game engine.
I did look though OpenJDK's license. It is a GPL project which means that anything linking to it needs to be GPL as well. However, they also have the classpath exception. I'm not sure if the linking portion applies because my code isn't linking to it any more than it would be with or without bundling it. As far as I know, the classpath exception doesn't cover this use case.
There is a similar question on Stack Overflow, however that refers to the situation where an application is developed with OpenJDK, I am referring to distributing with OpenJDK.