Over the past few weeks I've been working on a little react-native game, it's relatively simple and I have the game and the styling pretty much fleshed out.

I really do want to introduce a high score table, and some time soon I would like to publish my game on the App Store and Google Play. I've never published an app on either before, so I'm really looking for some advice on what solid options are for databases (preferably cheap or free) that would work for either platform.

I don't need to store much data, just the final score of the user and the name they entered. Any advice is welcome.