The Game Mechanic

It sounds like a frustrating job. Game mechanics, in the non-punny sense, are interesting indeed. I've always found it fascinating how some games can be deemed fun and others not, and arguably it take just one well executed mechanic to make or break a game. Think of Tetris without the ability to rotate blocks.
Miguel Sicart describes game mechanics to be "methods invoked by agents for interacting with the game world," and argues that there is a fuzzy line between what is a mechanic within the game and what is a rule. With a bit of leniency on the exact definition of a "game mechanic" I have a couple of examples from games I've found of what I though to be interesting mechanics.
Here is an excellent example of how a game with only a few well done mechanics makes an enjoyable puzzle experience. This free online game is called called The Company of Myself and implements a mechanic where you can create solid duplicates of yourself that follow your last path of motion. The game also has a friendly way of teaching users how to play.

The goal is simple: get to the green door.

This is an example of the much loved sub-genre of real time strategy games known as tower defence. The primary mechanic is the ability to plant towers on specific zones that automatically attack nearby enemies. An important mechanic that builds on this is that the towers behave differently. The variation in statistic and effects allows the strategy to enter the game as players decide the best places to put the different towers.

The enemies also had varying statistics, validating the range of different towers available
and heightening the strategy and need for careful planning.

This game had a particularly good reward system. The completion of each mission rewarded players with either one, two or three stars which can be spent on upgrades for your towers. This acts as an incentive to replay earlier missions to get higher scores once you've obtained better abilities.

There are two mechanics that define the Devil May Cry series; the ability to attack enemies with a multitude of different moves and a score system (arguably a 'dynamic') attached to these moves. These manifest in the aesthetic of a "style meter" that features in every game released to date. The style meter increases when different moves are linked together in quick succession, encouraging aggressive but intelligent play. The player is shown to be doing well by higher style marks and is given an overall higher grade at the end of missions which can go towards buying upgrades.

The latest Devil May Cry game made the style meter even easier to understand with real time visible additions to the players score and each addition reference something the player had done. This way players knew if they were doing well or not and could figure out ways to improve themselves.

The score mechanic is as old any game. A representation of the player's skill within a game brings with it many different things, as mentioned above, including competitiveness. Competition, with others and oneself, addresses a primal instinct to prove dominance over others and self-improvement. It feels good to be the best at something and game score mechanics make it easy to monitor how good you are at that particular game.

Upgrades have been mentioned several times already and on their own are key mechanics to many games. The ability to improve yourself in a way that can be monitored within the game gives players a reason to keep playing. The more upgrades they get the less difficult it will be to overcome future obstacles and achieve greater things. Sometimes these upgrades take the form of in-game statistics. It is a self-merit system.
Take Pokemon for example, where leveling up and evolving your pokemon gives them higher stats, making defeating opposing pokemon easier to achieve.

Many games have mechanics that provide players with numerical representations of their abilities within the game. Whether these numbers are low or high has in-game effects that makes the experience different depending on what each number represents.

The mechanic of being able to produce light from a torch in the Penumbra series contributed greatly to its atmosphere. The whole series was designed to provide an atmospheric experience that immersed players within the world, allowing the games to have an intense psychological effect. The use of a torch with limited batteries illuminated rooms well and provided comfort but also made players conservative of the torch's use. Either they brave the dark now and save for a more crucial moment or make use of the comfort it provides and risk their future mental state.