For me it was Misty in Red when I pick Charmander. Couldn't get past her, because Butterfree's Confusion isn't good against Starmie either. I had to go Vermillion to train him up to learn Slash. Then Brock in Yellow because of the typing difference again: I had to get a Mankey.
Whitney in GSC, as everyone has mentioned, and Clair as well, but not for the Kingdra. For the three Dragonair! They were a test in endurance.
Hoenn was Wattson when I picked Treecko, and Norman for the other two starters. Norman's two Slaking is another test of endurance..
Sinnoh didn't really have any difficult ones. I pretty much worked through all of them pretty well, even Candice in Platinum.
The great problem in Unova was Elesa, as everyone has said. Her Emolga and Zebstrika. Especially in BW2 where the Zebstrika knows Flame Charge. Fire and Electric moves take out most types you've got at that point in time.