I've had a few unexpected. Most involving shinies.
First Emerald file, running around on Rt 19(?), run into a shiny Linoone. I had no clue what a shiny Pokemon was back then. I caught it just cuz I thought it was weird. Still have it. On my white version. And now, every new emerald file I start, I end up with a shiny zigzagoon.
First HG file, was running through the grass in Rt 36, right before Ecruteak. I thought "Oh I won't run into any Pokemon.", and then BAM. Shiny Pidgey.
First SS file, going to try and catch a Larvitar at Mt. Silver. Outside, near the Pokemon Center, I run into a shiny Ponyta before I use any repels. Caught it. Use repels until I get where Larvitars are. Repel wears off, run into a Larvitar. Shiny. Two shinies in a row. Happiest moment of my life.
First White file. Breeding Joltik. I had five eggs in my part. Biking around for them to hatch, not really paying attention when they hatch. Third egg hatches.. I think.. "Huh. I thought its eyes were blue..eh I wasn't paying attention. Maybe I'm wrong." Go into party, it's shiny. Unfortunately it wasn't the nature I wanted. But it was Quirky.. so it more or less worked for me.
Now for non shiny moments:
Giovanni/Silver relationship. Never saw that coming.
Cobalion's random appearance in BW2
BW2 rival's starter came from an egg. Not Bianca.
Catching Rayquaza in SS and it's Modest nature with perfect Sp Atk.