Number One would certainly have to be the original Final Fantasy for the NES. I'd never heard of it before I stumbled across it the bargain bin at a local drug store. I always looked through the discounted games while my mom waited in line to get her prescription, then I would beg her to buy my top selection. On this day, after much caterwauling, she relented. I am very thankful. Probably the game that had the most impact for me on the NES.
Most recently, This War of Mine. I picked it up during a weak moment when it was on sale without any knowledge of it previously and now I have 80 hours into it.
Generally, though, I'm pretty careful about what I buy.