Most MCUs these days have sleep/stop/wakeup functionality. Basically, they can consume astonishingly small amounts of power while waiting on a trigger - usually

Most MCUs these days have sleep/stop/wakeup functionality. Basically, they can consume astonishingly small amounts of power while waiting on a trigger - usually on a GPIO pin - to wake it back up. Some MCUs also include a built-in realtime clock peripheral, often with options for running it off a separate backup power supply such as a coin cell. Those RTC peripherals can often serve the same 'wakeup' functionality without even using an external pin. So the device can setup its RTC, set an alarm for a week in the future, and then go into a very low-power sleep mode. For example, the STM32F030 or 031 would work, although their 32-bit 48MHz cores would be overkill for this application. But they are ultra-low-power ARM Cortex-M0+ which includes an onboard RealTime Clock peripheral and consumes like 5uA in deep sleep. I think TI's MSP432 lines are even more power efficient, but I don't know much about those chips.