Bugs is to the Looney Tunes universe what Batman is to DC- Not only the best character on his own, but the one with the most interesting Rogues Gallery. Elmer Fudd, Yosemite Sam, the Gremlin (one of the few to put Bugs in the pursuer role), Rocky and Mugsy, Marvin the Martian, Gossamer (the big, red, hairy monster), Daffy (Rabbit season!), Hugo the Abominable Snowman- the list goes on.
That said, I was always partial to Pepe. He really turned the standard Looney Tunes chase format on its head.