The Garrison Campaign

You get a Garrison Campaign quest every week on a rotation over 12 weeks. They require level 100 to show up. Garrison level is irrelevant. Each chain rewards 800 [currency=823] if you fill the assault bar.

- Darktide Roost -

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• Garrison Campaign: Darktide Roost/Garrison Campaign: Darktide Roost
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• Darktide Defender & Shredder Suppressor
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• Dirty Rats
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• Releasing Rylak & Downsizing the Darktide
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• Rylak Rescue (After completion you get [item=119130] worth 250 [currency=824] on your mail)
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- Ring of Blood -

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• Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood/Garrison Campaign: The Ring of Blood
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• How Tough Are You?
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• Entry Fee
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• The Champion of Blood (Also rewards [title=397] title)
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• Out of the Shadows/Out of the Shadows
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- The Sargerei -

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• Garrison Campaign: The Sargerei
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• Infiltrating the Sargerei
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• The Word of Socrethar + Shattering the Enemy + Pesky Podlings + An Ancient Threat
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• Facing Demons
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• Socrethar's Fury > Awards [item=119134]
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• Forging Ahead
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If you are wondering how the weekly quest spawning works exactly, I have done a lot of research into it. I have a lot of alts and for some reason I decided I wanted to get this done on all of them. Here are my conclusions.

Normally, you only get one quest per week. Sometimes, however, you get two quests the same week. This is guaranteed to happen for two of the quests:

• The same week you complete the last quest in the first set of quests you also get the first random quest in the second set (so you will get the 4th and 5th quests the same week).
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• It is also guaranteed when completing the second set of quests; when you complete your 8th quest you will always get Garrison Campaign: Deep Recon the same week.

You may also randomly get an extra quest the same week, but you will never get more than two quests the same week.

If you are eligible to get a second quest for the week, it will usually appear immediately after a relog. So if you want to get the whole chain done as fast as possible, relog and see if you have another quest waiting for you!

If do you have a quest waiting for you, it will usually show up on your map as a yellow ! over your Garrison. However, a few of them do not, so don't rely on that.

Quests appear weekly in the following order:

First Set
You get one or two of these four quests each week. The order is completely random, but you won't get any other quests until all four are completed:

• Garrison Campaign: The Sargerei (Reward: [item=119134])
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• Garrison Campaign: Darktide Roost (Reward: [item=119130] via mail)
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• Garrison Campaign: Seismic Matters (Reward: [object=234425] outside your Garrison)
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• Garrison Campaign: The Ring of Blood (Reward: [title=397] title)

Second Set
Works the same way as the first set. You get one or two of these each week and the order is random:
Note that you do not have to wait a week to get the first random quest

• Garrison Campaign: The Fall of Shattrath (No special reward)
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• Garrison Campaign: The Search for Owynn Graddock (Reward: [item=119151])
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• Garrison Campaign: The Broken Precipice (Reward: [item=119128] via mail)
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• Garrison Campaign: The Exarch's Call (Reward: [item=119144])

Final Set
From here, the final four quests are given in a linear order:

1. Garrison Campaign: Deep Recon (No special reward)
   Note that you do not have to wait a week to get this quest
2. 
   
3. Garrison Campaign: Crows In The Field (Reward: [item=119139])
4. 
   
5. Garrison Campaign: Every Rose Has Its Thorn (No special reward)
6. 
   
7. a. Garrison Campaign: Awakening (Reward: [follower=217])
   b. Primal Fury (Reward: [item=120311] and [item=120313])

Note that the bonus objectives that come with each set are completely optional - you don't have to complete them to progress. I recommend skipping them unless you need the extra Apexis Crystals!

Piggybacking off Tolaire's excellent comment here:
If you have stopped receiving or never received Garrison quests, there's a chance you have an open questline you haven't completed. Use the following macros to test for this- if you receive the reply "true" but don't have the corresponding achievement, it means you have open quests in that chain.

/run local out="Warlord's Council: "..tostring(IsQuestFlaggedCompleted(35841));print(out)
/run local out="Darktide Roost: "..tostring(IsQuestFlaggedCompleted(34450));print(out)
/run local out="Ring of Blood: "..tostring(IsQuestFlaggedCompleted(36289));print(out)
/run local out="Exarch's Call: "..tostring(IsQuestFlaggedCompleted(36169));print(out)
/run local out="Sargerei: "..tostring(IsQuestFlaggedCompleted(35196));print(out)
/run local out="Iron Siegeworks: "..tostring(IsQuestFlaggedCompleted(34033));print(out)
/run local out="Heart of the Breaker: "..tostring(IsQuestFlaggedCompleted(34410));print(out)
/run local out="Fall of Shattrath: "..tostring(IsQuestFlaggedCompleted(35685));print(out)
/run local out="Broken Precipice: "..tostring(IsQuestFlaggedCompleted(36117));print(out)

There are three Garrison Campaign missions that are not listed in WoWhead. I can't vouch for the veracity of these strings, but here's what I think they would be:
/run local out="Strike at the Heart: "..tostring(IsQuestFlaggedCompleted(36162));print(out)
/run local out="Bloodmaul Compound: "..tostring(IsQuestFlaggedCompleted(34289));print(out)
/run local out="Primal Fury: "..tostring(IsQuestFlaggedCompleted(36341));print(out)

Personally, I got false on every single one of these except the ones I had achievement credit for, so the mystery of the disappearing Garrison Campaign quests (4 weeks without a mission and counting) continues.

::edit:: if you are still stuck on the Garrison Campaign, open a GM ticket. They will force a quest to spawn.