I'm creating a game with a reasonably long story, chapter by chapter. This means the full story hasn't been created yet. Can I release a game like this to the Google Play store & add content later, or does it have to be completed?

Most importantly can I charge for an unfinished game? (On Google play you can't change a free game to a paid one, it has to be paid from the beginning).

You can update a game on Google Play as often as you like - so if you want to amend the game etc. you can do so.

As long as your description is clear i.e. here are the first 3 chapters / levels and more will be released as they're completed, users should be okay with that. 

 If you don't specify that, then they'll most likely ask for a refund as they might feel they paid for something they didn't get. 

Another solution would be to have in-app purchase and you could release them in blocks like:

• Chapter 1 or Level 1 is free
• and then charge $1 for every additional chapter, or every additional 2 chapters depending on your price-point.  That way you're free to add in chapters, and people can buy them or not as they wish and they won't feel that they've paid for something they haven't received.

Or split your project into versions :

•  LongStory 1 - is the first 5 chapters or levels that they buy separately
• LongStory 2 - is the next 5 chapters or levels that they buy separately etc.

Google Play does not have a definition of what counts as a “Complete game”. So yes, you are allowed to submit an unfinished project. However keep the following in mind:

• Google Play has support for Alpha and Beta versions of an app. These might work in your case, as users will have the expectations that the project is not finished yet, although I’m not sure how it affects visibility.
• Make sure you are perfectly clear to potential customers what your app does and does not include. The last thing you want is negative reviews from people expecting a full game, before it is fully released.
• As you noted yourself, you need to charge from the very beginning.

I’ve used Google Play a lot a year or two ago, not sure how much things have changed since then, but I’m fairly positive it’s still (roughly) the same.

I've published games where I released updates with additional levels, so I don't see that a story-based game is necessarily different. However, I'd avoid describing the game as unfinished in any way (maybe "episodic" or "continuing" would present it more as a feature). Aside from the marketing aspect, it would be less risky, rejection-wise. I once got an App Store rejection because the description had the word "sample" in it, even though I was referring to another project that I was crediting. Google Play may be less strict, but in my experience even there paid apps seem to get more review scrutiny than free ones, and when I've had a rejection it was a real hassle figuring out exactly what their objection was and even how to respond.

Thanks guys these are great responses! It sounds like what I want to do is allowed, as long as I'm upfront & don't try to deceive anyone (which I wouldn't want to!), so I'll try to make it as clear as possible. 

Thanks!