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The Android and iOS versions of stellarium are not open source so you can't compile you're own versions.

But can I use it to build a better, paid-for, app without having to give up my source code because of the GPL thing?

Not allowed to install, no. Can’t recommend it.

As I said, don’t mind here and there… it’s helpful. But nag me and get replaced.

Its worth mentioning that nobody is doing anybody a favor by using their free software. It seems pretty plain that the free version is a trial you are expected to uninstall or upgrade.