Cave Explorer Outfit - You require 100 fame points in Spike Tasks Quest to complete the full outfit, which will take a minimum of 25 days.
I think this outfit can take the longest time. Of course, there are outfits that need ingredients to get them, such as Mage Outfit, Summoner Outfit, but it's the matter of money and not time that you need to invest in it, you can get them in no time, and for Cave Explorer you need to make tasks daily.