Question: The documentation says it should be used instead of LocalStorage, but it doesn't list anything regarding how large it is. If I recall, LocalStorage in web browsers is only about 10mb.

<https://facebook.github.io/react-native/docs/asyncstorage.html>

Answer: 
It writes to disk and is very large if there is free space. On iOS there are no artificial caps other than what the underlying operating system imposes: <https://stackoverflow.com/a/1897383/454967>.
On Android the limit is 6mb by default. Check [this](https://github.com/facebook/react-native/blob/master/ReactAndroid/src/main/java/com/facebook/react/modules/storage/ReactDatabaseSupplier.java#L46). 

But you can increase that limit calling the method [setMaximumSize](https://github.com/facebook/react-native/blob/master/ReactAndroid/src/main/java/com/facebook/react/modules/storage/ReactDatabaseSupplier.java#L142).