Or just reporpose an old phone.
Even with that, a clock generally runs for months, even years, on a single battery. Not exactly phone territory, but might be feasible on the ESP8266, depending on how the oscillator behaves in deep sleep mode.
horribly
TI's bq32000 is $0.55@1K, is 3.3V for easy interface to the ESP, and takes just over 1 microamp in backup power mode. That's just one that I quickly poked at.
(Side note: at this point, I'd think I'd want to be using an ESP32 in any new designs.)