I have two that I'll share.
We bought our first mare from a well known Arabian stable in the area. She had been bred, but we didn't think she'd caught because she appeared to come back into heat. This was in the early '90's, so I'm not sure what US technology was at that time for horses, but she didn't get an US. It was late in the year, so we didn't breed back. Anyway, we were pretty young kids and didn't ride her alot throughout the cold winter months and so when she appeared to "put on a little weight", we thought nothing of it. Even as Spring wore on, she looked a little bit chubby, but again, with two little kids riding her, we didn't necessarily provide heavy exercise. Then, seemingly overnight, her belly dropped and was HUGE and she had a massive bag. My mom called the vet out and sure enough, she was pregnant, and based on her last breeding date, due any day. Unfortunately, there were complications during labour and we lost the foal, but I won't get into that. The whole thing was funny though, because she literally showed nothing until just before her foaling date.
The other one I have was with my mare. She was due early July. I was a teenager and was out every single morning, first thing, rain or shine, eagerly waiting for the expected foal. Every morning, nothing. A friend of mine who lived in the city invited me to stay for the weekend. I thought, "sure, why not, she's never going to have this thing." Sure enough, I got a phone call the very next morning from my dad, that the foal had been born. The worst part, I wasn't going to be leaving my friend's place until the next afternoon!!! Longest. Weekend. Ever.