I have had two successful rare encounters and 12 fails (I no luck). Here it is in order.
1. FAILED Dragonite. On day 1 a dragonite popped up on the nearby. I had no clue what it was, so I just remembered it and what it was. Kind of irritating to think back to then and think that on day 1 rather than a kingler being my best catch a DRAGONITE.
2. FAILED Muk. Me and my cousin were in my house when a Muk appeared on the footsteps. We searched for it, but we missed it.
3. ANOTHER FAILED Dragonite. This time I knew what it was and I got really excited. I literally ran all round my neighbourhood trying to find it, but never did.
4. FAILED Vaporeon. This was after i literally got 24 flareons in a row from eevee evolutions. My best pokemon was flareon and 9 of my top 12 were flareon. It disappeared almost immediately once I noticed it.
5. FAILED Blastoise. Spawned at my house, ran after 1 ball - it was even more annoying because it was ??? CP and was my last candy for wartortle.
6. SUCCESS Lapras. A lapras appears on my nearby while I am eating and I run out to get it. Capture success.
7 8 and 9. FAILED Snorlax.
Day 1: In the city and a snorlax pops up on the nearby but we can't find it.
Day 2 Snorlax pops up at a nearby park. Catch it but CP=44
Day 3 Snorlax appears again in the same place this time it has the amazing cp of 42.
10 FAILED Tyranitar. A few days have passed of GEN 2. As I am walking I see a Tyranitar at the next pokestop. All of a sudden I see the message of disappointment: AN UPDATE IS AVAILABLE. WOULD YOU LIKE TO UPDATE NOW. And I have no space. Probably the worst one especially because if the update could have just waited 5 minutes we would have been done here.
11 FAILED Unown. A wild (non event) L unown (lol it gave me the L from the start) appears on fast poke map. I get down there as fast as I can, and it appears. Tapped it. The game says "Error" and the unown is gone. Rip
12 FAILED Tyranitar. A Tyranitar appears at the local park. And of course it runs immediately.
13 SUCCESS Dragonite. Dragonite appears straight after a group of 5 of us fail a charizard raid. We all catch it.
This is obviously not all - there was a wild charizard which was the last poke I caught in 2016. I just didn't want this too be too long. There have been 3 more Tyranitar fails. 3 FAILED togetic, a FAILED ampharos and about 2.7 quadrillion FAILED magikarp. (Okay maybe that last one was a bit overexaggerated - I doubt I have FAILED 2,700,000,000,000,000 magikarp.