I have a Garmin Forerunner which satisfies these requirements. It's basically the size of a normal watch, notifications work very well, and it has a GPS for tracking runs if you want that. I've had it for years and charge it at most once a day for ~15 minutes when I'm in the shower and it has literally never run out of battery. You can take it on a weeklong camping trip and probably be fine as long as you don't use the GPS.