examples: Update `basic` example to Next.js 15. (#9440)
### Description
As the title says, updating to Next.js 15 in `basic` so `create-turbo`
gives people the latest and greatest.
I've also taken the time to:
- Make the content of the applications Turborepo
- Added a `check-types` task for folks who are interested in running
TypeScript checks.
### Testing Instructions
Our examples CI should make sure things are still building correctly.
Would appreciate if someone could open it up in their editor and see if
they get any squiggly lines, as well!
