Boy You're So Lucky....

*Legendary pokemon wore off at 50% HP or below*
*throw ultra ball*
*ball shakes* C'mon....
*Pokemon got lose* dang it!

*Opposing pokemon use powerful move, but misses* That was close
*My pokemon below the opposing level use a Super Effective Move, lands critical hit* Wow. I'm lucky

Well I can't think of many right now. You'll be glad opposing pokemon couldn't land a hydro pump or fire blast and you manage landing their hp at such puny level with critical hits. and the time some decent balls didn't help out and failed at catching.

The thing that we all have to carefully consider, damage rolling in Pokemon since Gen 1 is random. This is something different from Critical Hits.

I remember those moments where I got KO'd not because of a critical hit but its because Damage rolls hated me and decided to let the AI get the 100% Damage Roll (When i mean 100%, the maximum possible amount of damage). Max damage rolls + Critical hit is something that makes this game so surprising at times.

Its even more hilarious when it happens for the player's favor. I had a moment of grace in Pokemon Y with the battle against Diantha when I used a Max Revive on my Lucario because i thought that I'm basically in the inevitable loss against her, with Mega Gardevoir wiping out most of my remaining party members before i used the Max Revive. So my max revived Lucario with Shadow Claw went up against the inevitable, Mega Gardevoir who could clearly defeat it with whatever STAB Move it had (Moonblast or Psychic, both of which will deal HIGH amounts of Neutral damage against the Fighting/Steel type either way) but i suddenly rolled a CRIT AND THE MAX DAMAGE ROLL on Mega Gardevoir, on a NON-STAB Shadow Claw and scored the KO on her, which i honestly thought would never happen. (The most that my Lucario at Lvl58 did to Diantha's Mega Gardevoir (lvl68) did most of the time was around only 80% of her Health or less).

Then there's also my moment with Battle Chatelaine Evelyn in AS who for some reason, her Primeape's Stone Edge was dealing disgusting amounts of damage to the point where it was OHKO's things without Stone Edge critting, and again, NOT ON STAB. Again, Damage Rolls...

I think the best surprising moments are when you land a side-effect for a given move.

As I've said in other threads, the best/worse part of that is Poison Sting.

When you're in trouble and you're gonna die an your opponent also, and then you poison it and it dies without getting a chance to kill ya.
But the reverse scenario is awful. I'm sure guys have been KO'd at least once by a Weedle. Weedle.

Chance and luck are part of the game. Without it the whole game would be completely predictable, would be boring, wouldn't it?

I'd have to second above; although it may seem unfair to some, especially when battling competitively (in which case, if your strategy doesn't go well enough due to your opponent's positive luck, it'd lead to frustration or disappointment), it helps make the system more unpredictable and whatnot, and I'm sure a lot of us appreciate that (even if we may not realize it).

I think luck is both a positive and negative aspect of the game, because it's fun when you're the one rolling the dice but then it's not so fun if an unexpected critical hit or status hax screws you over.

I know many situations where I lost battles on Showdown thanks to hax and critical hits, but there were also ones where I was the one throwing out the hax and sometimes it saved me. But it's definitely not fun when your move misses right at the most critical moments.

As H.A.N. said, without luck I think the game would be really boring and predictable.

There were a few times I caught legendary Pokemon in ordinary Poke Balls: Moltres in LeafGreen and Virizion in Black 2.

In terms of other battling aspects, I can't think of any other than losing in battling facilities because of moves missing, mainly because of computer hax.

I remember reading in a Nintendo Power article once (geez, I'm really dating myself here) about luck in Pokemon Red/Blue games. The writer mentioned that some times outcomes might feel unfair, but there's plenty you can do to stack the deck in your favor. When I stopped to think about it, calculated risks kinda add an extra layer of excitement to the game you wouldn't get from, say, a chess match. Since then, my favorite strategies involve risky maneuvers that are more thought-out than you'd expect. It's always more fun to come out on top when it's done in an interesting or unorthodox way.