I guess this all depends on what Pokemon are in your party, but personally, there were three battles I found particularly hard (I played on Normal difficulty). Feel free to share your experiences too.
1. Melia Battle on the ship
I don't know why this battle gave me so much trouble, but I really didn't have anything for her Gardevoir and Yanmega, and hence they swept my team a couple of times. On my second run, I had a Crobat so her Yanmega wasn't a problem, but her Gardevoir was still a massive threat.
2. Saki & Venam Double Battle
This wasn't as difficult when I faced them on my second run, but ALL of the girls' Pokemon hits hard and are bulky at the same time (especially that goddamn Metagross). Surprisingly, Melia was pretty helpful on my second run (she was useless on my first run. She kept spamming Flamethrower on Dragalge when there was a Metagross right next to it). I'll try to take them on by myself on my next run - but I bet it's going to take a long time...
3. Geara & Zetta Double Battle
(Just realized that all three battles involve Melia to some extent *shudders*)
This would've been easier if Melia hadn't brought only three Pokemon. Geara and Zetta seemed to also like focusing their attack on one Pokemon, meaning that a Pokemon on our side was most likely to faint on that turn. Due to the field, Water and Ice type moves are heavily weakened, so that doesn't really help either (I hate Zetta's Gigalith with a passion).
Honorable Mentions:
- That random trainer with a Level 85 Alakazam in Gearen City. WTF
- Madame X