I've created unique set of quests as an alternate way to summon Daedric princes. When arriving at a dungeon you may receive a quest to explore every dungeon of that type in the region. Completing enough quests of the same type will allow you summon the associated Daedric prince without any chance of failure. My goal is to give you reason to explore each region and conquer far more dungeons. There are 91 of these quests!

Here are the associations and the number of dungeons needed to summon:

Crypts Vaernima 30
Orc Stronghold Boethiah 24
Human Stronghold Clavicus Vile 25
Prisons Mehrunes  21
Desecrated Temples Azura 27
Natural Cave Meridia 29
Covens Nocturnal 29
Vampire Haunts Mephala 29
Laboratories Hermaeus Mora 31
Harpy Nests Hircine 20
Ruined Castles Namira 36
Spider Nests Malacath 30
Giant StrongHolds Molag Bal 30
Dragon Dens Peryite 29
Barbarian Stronghold Sanguine  20
Volcanic Caves Peryite 29
Scorpion Nests Malacath 30

You can only do one quest at a time. There is no time limit.
You will receive the quest when arriving at a location.
You will not receive the quest if you already have a dungeon of the same type assigned to another quest.
You will automatically receive the quest to summon the Daedric prince once you reach the target number.
If you somehow cancel the quest to summon a Daedric prince, it can be restarted by going to a cult shrine.

You will also receive 15 legal rep in that region and 1000-2000 gold per dungeon for completing the quest.

Currently, you will also be able to do the Daedric summoning the normal way as well in order to get 2 rewards, but this will likely be changed in the future.

If you use Smaller Dungegons, Fixed Dungeon Quest Markers is required to ensure you get the correct list of dungeons.

Here is the full quest list:

Alik'r Crypts
Alik'r Desecrated Temples
Alik'r Nature Caves
Alik'r Ruined Castles
Alik'r Scorpion Nests
Dragontail Mountains Crypts
Dragontail Mountains Orc Strongholds
Dragontail Mountains Human Strongholds
Dragontail Mountains Natural Caves
Dragontail Mountains Volcanic Caves
Dwynnen Crypts
Dwynnen Prisons
Dwynnen Desecrated Temples
Dwynnen Vampire Haunts
Dwynnen Laboratories
Dwynnen Ruined Castles
Dak'fron Crypts
Dak'fron Natural Caves
Dak'fron Dragon Dens
Dak'fron Scorpion Nests
Wrothgarian Mountains Crypts
Wrothgarian Mountains Orc Strongholds
Wrothgarian Mountains Natural Caves
Wrothgarian Mountains Vampire Haunts
Wrothgarian Mountains Laboratories
Wrothgarian Mountains Giant StrongHolds
Wrothgarian Mountains Barbarian Strongholds
Wrothgarian MountainsVolcanic Caves
Daggerfall Crypts
Daggerfall Prisons
Daggerfall Covens
Daggerfall Vampire Haunts
Daggerfall Laboratories
Daggerfall Ruined Castles
Daggerfall Giant StrongHolds
Daggerfall Dragon Dens
Glenpoint Crypts
Glenpoint Desecrated Temples
Glenpoint Covens
Glenpoint Vampire Haunts
Glenpoint Laboratories
Sentinel Natural Caves
Sentinel Covens
Sentinel Ruined Castles
Sentinel Scorpion Nests
Anticlere Crypts
Anticlere Human Strongholds
Anticlere Desecrated Temples
Anticlere Natural Caves
Anticlere Covens
Anticlere Vampire Haunts
Anticlere Ruined Castles
Wayrest Orc Strongholds
Wayrest Desecrated Temples
Wayrest Covens
Wayrest Vampire Haunts
Wayrest Ruined Castles
Orsinium Orc Strongholds
Northmoor Covens
Northmoor Harpy Nests
Ykalon Natural Caves
Ykalon Laboratories
Ykalon Ruined Castles
Ykalon Spider Nests
Ykalon Barbarian Strongholds
Abibon-Gora Harpy Nests
Tigonus Harpy Nests
Tigonus Scorpion Nests
Totambu Crypts
Totambu Orc Strongholds
Totambu Human Strongholds
Totambu Desecrated Temples
Totambu Covens
Totambu Vampire Haunts
Totambu Laboratories
Totambu Ruined Castles
Totambu Spider Nests
Totambu Giant StrongHolds
Totambu Scorpion Nests
Ilessan Hills Crypts
Ilessan Hills Human Strongholds
Ilessan Hills Prisons
Ilessan Hills Desecrated Temples
Ilessan Hills Natural Caves
Ilessan Hills Covens
Ilessan Hills Vampire Haunts
Ilessan Hills Laboratories
Ilessan Hills Ruined Castles
Ilessan Hills Spider Nests
Ilessan Hills Giant StrongHolds
Ilessan Hills Barbarian Strongholds