Repo: Add a way to detect breakages from new TS versions sooner

[JoshuaKGoldberg]

Suggestion

Splitting out from microsoft/typescript-error-deltas#119 (comment), courtesy of #7088: is there a way we can be informed of issues from new Typescript packages sooner?

I'd love to get to a point where we always spin up a feature branch whenever there's an RC/beta, but historically it's been hard to find maintainer time consistently...

[jakebailey]

Effectively, I was just thinking about adding to your CI a nightly scheduled run, plus an if-'d task which installs typescript@next. That way, the nightly run will fail and give you some early warning.

Something kinda like what I do in hereby to make sure that if my package is installed without a lockfile that things work:

https://github.com/jakebailey/hereby/blob/d9b8347923e3b098d00affdc4baa725aff00f9db/.github/workflows/ci.yml#L33-L38

[bradzacher]

Maybe something like #1752 would help? I.e. once we can test old versions - it seems logical we could use the same infra to install the nightly TS version instead and run it (perhaps in a job that runs weekly or something to reduce noise?)

I remember we also talked about having a job that ran regularly against the latest tag for both ESLint and TS to ensure nothing breaks on the newer version (because we don't always update with minor ESLint versions or TS patch versions). I can't find an issue for it but I remember it... Something vaguely on that line is mentioned in #6456