You've done a great job demoing a variety of patterns and tools. I'll add two more things you could benefit from learning early:

- [Type](https://docs.python.org/3/library/typing.html) [hints](https://stackoverflow.com/a/32558710/10135377). Function headers are almost always clearer with type hints, and you need them if you want to use a tool like [MyPy](https://github.com/python/mypy).
- [The main method pattern](https://stackoverflow.com/a/22493194/10135377).
- (Usually I'd also advocate for recursion when someone's demo project is a REPL game like this, but in your case I don't think it would improve this code.)

Other stuff:

- Usually it's better to avoid exceptions when you have other options. While you could do what you're doing here by other means, you've done a few things that make me like your `check` function: It's concise, it bundles away a discrete bit of functionality, and you're catching a narrowly defined exception class.
- Unfortunately, it's a little clunky to have a function that just tells you if `int(x)` threw an exception, just so you can call `int(x)` on the next line. You have many other options; I'd be fine with `def check (user_input: str) -> Optional[int]:...`, but then you _must_ use `if x is [not] None:...` later. 
- Depending exactly what the function in question does, either `validate` or `sanitize` would probably be better than `check`.
- I lied: another new thing to learn: [itertools](https://docs.python.org/3/library/itertools.html). In particular, a `while` loop that increments something is always begging to get replaced with a `for` loop. In this case, since we want to keep going "as long as it takes", we need a lazy infinite iteratable: `itertools.count(0)`.
  - In order to make that work, we'll need to separate out the "that input was invalid, try again" logic into a separate loop (or recursive function). 
  - And then if you teach yourself [generators](https://wiki.python.org/moin/Generators) you could write `for (try_number, user_input) in zip(itertools.count(0), yielded_user_input()):...`. Fun times!