Repeatable Quest

While many of the quests in games that use them are one-off affairs, this type can be repeated with the same NPCs, objectives and rewards. Some can be repeated indefinitely and consecutively, allowing a player to accumulate several stacks of the necessary items (by hunting or shopping) to hand in at once, while others may have a 'cooldown' or a limited number of repeats.

Repeatable quests allow players to obtain the precious chunk of extra Experience Points and cash that usually comes with a quest more than once, which may allow them to offset losses from deaths. Particularly rewarding repeatables enable difficult and tedious 'main' quests to be skipped by easing the Level Grinding needed to enter the next area. Repeatables can also be designed to supplement said main quests if both have similar or identical objectives, allowing players to squeeze some extra profit from their first playthrough of an area.

A number of games with repeatables present them as the day-to-day chores of NPCs or factions, allowing players to gain their favor by essentially doing their menial work. Savvier writers can thus flesh out the involved characters through the quest descriptions, which players who stick to the main storyline will not see. Caution is needed as making too many repeatables urgent or assigning objectives that are too grandiose can shatter Willing Suspension of Disbelief. The use of set pieces is a double edged sword in this area as while they can make the outcome of a repeatable spectacular they also force events to play out in a similar way each time. Finally, there are players who may find that being asked to do NPCs' odd jobs for them is simply demeaning.

At its core, this type of quest is just another way of dressing up the Level Grinding prevalent in many games and more jaded gamers may consider it to be no improvement. However, developers can add appeal to repeatables by providing multiple options from a pool or allowing certain characteristics to change over time. The fact that there is an additional reward at the end can be a psychological draw and of course, good writing and a careful choice of objectives can make the quests entertaining.

Due to the simplicity of their objectives and the ease with which they can be 'meshed' with core story quests and each other, the Fetch Quest and the Mass Monster Slaughter Sidequest are often made repeatable.

Do note that most repeatable maps, levels or encounters that enable farming do not fall under this trope but are a near omnipresent gameplay convention. Repeatable Quests must be:

QUESTS that follow the stereotypical format of doing something for a NPC.
REPEATABLE in that the NPCs and tasks are identical for every iteration or change according to a fixed pattern. Randomly generated minor quests do not count unless there is some way to deliberately or accidentally lock combinations of quest characteristics.

Examples of Repeatable Quest include:

MMORPGs

World of Warcraft does this with 'dailies' and may even have been the Trope Codifier.
- Eden Eternal uses a very similar system except with shorter cooldowns on the quests.
- Dailies are also present in Iris Online.
Dragon Saga pads the levels between the 'big' main quests with various repeatables. Notably, the Infinity+1 Sword crafting quests are almost all repeatable and of the unlimited variety.
Maple Story did not introduce these until later versions despite growing and rather vocal demand for them in the early days.
Earth Eternal is big on these.
Mabinogi allows the players to perform part time jobs with the game's vendors, and allows them one in-game day to complete them. This is a useful means of accruing monetary wealth, especially for characters who are primarily crafters.
Shin Megami Tensei Imagine has a small variety of these, including quests you can only repeat if you pick up uncommon loot.
Ragnarok Online and its Bragging Rights Reward hats (among other awards, but mostly hats) can be obtained in Fetch Quests, and can be repeated as long as the player is willing to hunt down the same items.
In Guild Wars, many of the quests offered at the annual "Wintersday" festival (which falls around the time of Christmas-New Year's) are repeatable, enabling players to farm huge amounts of holiday items.
Online play in the Monster Hunter series consists almost entirely of repeatable hunts, so the quests are naturally repeatable as well. However, the extended preparation required for each task keeps players involved.
Final Fantasy XI has numerous repeatable quests ranging from the simple Fetch Quest for fame grinding to redoing major battles in Missions for a different end reward.
Dungeons and Dragons Online adventures are infinitely repeatable on four different difficulty levels each. The first play on each adventure/difficulty combo gives an array of experience bonuses, while subsequent replays start adding penalties, eventually hard-capping the XP gain at a very low number for extreme repetition. Ironically, it's more efficient to not repeat quests too heavily (except for crafting purposes).
Dragon Nest has a different sequence of optional quests for every mission map but the last quest in the sequence is always an infinitely repeatable 'Assault' quest that consists solely of finishing the map on the second hardest difficulty. Accepting this quest repeatedly and rushing through the map each time forms the basis of most organized Level Grinding.
Rift has its share, just like the World of Warcraft ones.

Single-player games and RPGs

In Majora's Mask, all sidequests are repeatable in a manner of speaking as the game's three day cycle allows them to be retaken during each iteration.
Quests in the Monster Hunter series tend to be repeatable offline as well as online. It would be hard to get all those rare materials otherwise.
Final Fantasy XII has only one repeatable quest, and it's very modest, and available very early. By visiting the Giza village and running around in the Giza Plains area, you can create multiple Sunstones, which sell for what is a moderate bonus for the player at that part of the game—but the player could get the amount from just killing wolves and selling the loot, and on the side get both EXP and LP.
In Mega Man ZX, Rose and Huguet will request you to go to Area Lin order to gather some supplies for them. After going to three diferent sections each time, Rose will give you a sub tank. However, you can only obtain two sub-tanks for this and any subsequent attempt only nets a token thank you and a few Energy Crystals.
Many of the quests in Puzzle Quest are repeatable, such as an early quest to patrol the area for monsters.
Several of the multitude of sidequests in Dragon Quest IX are repeatable, though sometimes the reward is reduced after the first completion.