Mount Waialeale, Hawaii records 360 rainy days per year:
https://en.wikipedia.org/wiki/Mount_Waialeale#Climate
López de Micay, Colombia at 315 days:
https://en.wikipedia.org/wiki/L%C3%B3pez_de_Micay
Mount Read, Tasmania at 282 days:
https://en.wikipedia.org/wiki/Mount_...mania)#Climate
But I'm pretty sure there are places with more rainy days (such as Mawsynram in India, though it doesn't measure rainy days).