The battle against the leader Whitney in HGSS was really a killer for me. Her Clefairy kept using Sing on my Pokémon, and put my whole team asleep in no time at all. After repeatedly using Flash and many Awakenings, I managed to get in enough hits to kill it off. Then, oh boy, her Miltank. It would keep using Rollout, killing off my team slowly, and after a lot of Potions and sending Pokémon in and out, I had been close to defeating it... only to have it use Milk Drink, which restored it's HP almost completely. I got defeated by her at least three times. The fourth time, I had grind trained my Quilava and Noctowl, but when it came time for Miltank again, I almost lost... except, by pure luck, I landed a critical hit with my Quilava;s Rock Smash. I was so shocked and relieved that I cried.
Then there was the champion battle against Iris in B2/W2. I plowed through most of her team somewhat easily with my Mamoswine's Ice Beam, but when it came to her Hydreigon and Haxorus, their Flamethrower moves took Mammy out of the picture, then they quickly bowled over the rest of my team, since I really wasn't expecting her team of all Dragon-Types, and since I didn't have any other Pokémon super effective against them, I was done for. After five attempts, I gave up and started to grind train my Mamoswine, Lucario, and the weaker Pokémon on my team, until I was stronger and feeling confident again. Along the way, I had actually discovered for the first time that Lucario could learn Dragon Pulse, so when I had taught it to him using a TM, I felt ready for battling Iris once more. I plowed through the Elite Four, took a deep breath, and faced Iris. With my highly trained team, Mammy's newly taught Blizzard, Naruto's Dragon Pulse, and lots of extra help from my other Pokémon and many Full Restores and Revives, I managed to defeat Iris. I felt so proud of myself.
This doesn't really count as a battle, but I was crushed by the two Rhampardos in Pokémon Ranger; Shadows of Almia. Their Rock Slides kept breaking my Capture Stylus' string (I think that's what it's called; my memory is a bit fuzzy). I actually gave up on the whole game for months until I decided to go back and retry. I got defeated again and again, so I then went and raised my ranger's level a lot, and it took me days to get to a point where my level was high enough to finally capture the two devils, and it was still a very close call.