First off, fishing does not count for this exercise. Second, the minigame cannot require the player to have some minimum number of levels within the MMO, unless it's a really low number. Third, the minigame must reward the player with game gold or items. A good example of what I am asking for: the MMO Wizard 101 has a fairgrounds in the starter city where any player can play several minigames: a columns clone, a dig dug clone, a concentration clone, etc. The standard reward of playing is to refill the player's health and magic meters, but the game also rewards money and items. An interesting but borderline example: the game Neopets is only sort-of an MMO, it has no world for the player to walk around in and kill monsters. But it has a huge collection of flash games which are the main source of players' daily income. The player can submit three scores per game per day, and be paid according to how high the score was. Games usually do not reward items, but a few of the games reward food items for feeding the pets (this is quite parallel to health and mana pots).

Guild Wars has some nice minigames, like Rollerbeetle racing, Snowball Arena and Polymock.