I'll ignore the obvious ones like Brock's flirting or TR's motto, but I'd say some of the others that went on too long:
- James always having a grass pokemon that attacks him. I thought it was kinda cute when Cacnea did it too but by the time Carnivine arrived I was done with it.
- Jigglypuff was heavily overused in the early seasons, to the point where it appeared almost every 2-3 episodes until Johto started.
- Sukizo's "Remarkable" during the Pokemon Contests.
- Pikachu getting sick when it stores up too much electricity, they seriously repeated this in like 5 different episodes, even with Pachirisu.