In League, I tend to play a number of lanes and, naturally, am often forced to choose my champion before I know who is on the other side (either because it's blind or I'm higher in the pick order in draft).
On top, one of my favorite champions is Fiora (I'm sure someone here can guess why). From time to time I'll pick her only to find I'm forced to go against singed. For a game with relatively few hard counters, there isn't a much harder counter than Singed versus Fiora.
Mid I tend to play a handful of champions (Lux, Twisted Fate, Morgana). As either Lux or Twisted Fate, if the other guy picks Kassadin my response is usually to ask the ADC to switch lanes - not a lot you can do once you both hit level 6 and Kassadin can instagib you. I don't really have that as Morgana since it's possible to play very efficiently, constantly harrass kassadin, and protect from his silence - plus my build for her tends to include a lot of HP (Liandrys and Rod of Ages) so his damage isn't quite so intimidating if I make a mistake.
Beyond League, I've rarely played games where counters exist. Smash Brothers Brawl had Marth who was unreasonably powerful thanks to his long range, quick attack speed and ability to overpower the vast majority of attacks with his own.