Final Fantasy IX: At the beginning of the game, answer Baku’s question wrong 64 times for a slightly funny bit.
Final Fantasy X: Dodge lightning bolts two hundred times in a row in the Thunder Plains to get the Venus Sigil for Lulu
Kingdom Hearts: Enter and exit Geppetto’s house twenty times (more or less) and then talk to Pinocchio and he gives you the "Chocobo" gummi block.
Tomb Raider, Various Versions: The famous "wallsy" bug — go up against the edge of a wall and jump repeatedly at just the correct angle to access secret items or explore the roof of Lara Croft’s house.
In a blatant example of corporate propaganda, in Wonderboy on the Master System (an oldie I know), collecting all four letters of the Sega logo hidden within various stages gave you an extra life.
Another is in King of Fighters: Maximum Impact. Fighting the final boss Duke with 300% Recovering Health. ;_;