With the Press your luck mechanic another mechanic is repeated until the player chooses to stop, and if they continue past a set point then they lose what would have been acquired. For example in a game with the Auction mechanic each player continues to bid until all but one drop out and the 'winner' of that auction gets the thing they were bidding on. Using the straight Bidding mechanic players typically secretly choose an amount of a resource to bid on the item and the 'winning' bid gets that item. Another example would be games where players have to in order flip from a deck and collect all cards flipped, if they choose to stop before a pair is revealed. This second example is typically paired with Hand management/Set collecting.
An example from the library's current collection is the game For $ale. Another, very different game, using this mechanic is Formula D. In this game it is a secondary mechanic to Role and move, but the player can choose their gear and therefore the range of numbers on the die that they role. The critical thinking and problem solving come into play when trying to decide of this gear, because as the car goes around the track it can be damaged and loose you a chance at the race victory!