Wow, this is actually a very deep and engaging question. It's not a sliver of memory for me.
The first time I was captivated by The Amazing Race was during my senior year of high school - class of 2002. It was close to spring break and I was quite a lone ranger back then. I had a couple of acquaintances to call "pals", but we never hung out outside of the classroom so those friendships became duds after graduation. My parents were a bit strict. I didn't own a phone because I couldn't afford a phone plan out of my allowance, nor was I allowed to use the landline unsupervised; thus, I was basically stuck at home and I could only mow lawns and do some gardening if I wanted to earn money and go out. I was particularly bored this one evening after doing garden work and crying from people not wanting me around them. I switched on the TV and lucky enough, I landed on CBS where I saw two girls rowing a raft with giant sticks. I was sold on watching the show when I saw them giving a bath to an elephant because who doesn't love elephants?! And best of all, they took the entire leg home with a win! I would later find out this was Mary & Peach during Season 2's Chiang Mai leg. I watched the following weeks and had a roller coaster ride of excitement that I would otherwise would have missed out on, and replaced my hurt from being a typical high school outcast. I fell in love with seeing the sights of Hong Kong and I LOVED Oswald & Danny after they chose to go shopping in Hong Kong. The Australian slang also won me over after I initially believed that the language was a myth. I was hooked since! Watched during college (and made better friends), made it home from the actual workplace in time to watch the show (and with wonderful co-workers), and life actually got better because of TAR. It's somewhat silly, but I call this show my good luck charm. Certainly still way different than anything on TV now and helped me through my young adulthood.