Well (expressing hesitation, etc.) ... this "extreme weather" is a bit general. I understand it as "life-threatening weather" Let's give a unified answer.

ROG:

Winter (low temperature, frostbite): 15 days

Summer (high temperature, heatstroke, spontaneous combustion): 15 days.

Software:

Hurricane season (hail, gale): 18 days

Monsoon season (waterlogging, rain, poisonous mosquitoes): 18 days.

Volcanic season (high temperature, meteorite): 17 days

See the "season" entry in "Don't starve" wiki for details and solutions.