Bog Beast

Here's how to extend the quest for as long as it's necessary for the rare to spawn without joining a raid group. While Muck It Up is available:

1. 
   
2. Collect 4 [item=177880].
3. 
   
4. Toss no more than 3 out of 4 towards the objective. Important: if you toss all 4, you will no longer be able to loot [item=177880].
5. 
   
6. Delete the rest.
7. 
   
8. Just like the warning says the quest will be abandoned, hence reset, and picked up again after a short delay.
9. 
   
10. Rinse and repeat.
11. 
    

It took me 5 resets to make the rare spawn.

Good luck on the pet!

Bog Beast is located in the Revendreth zone.
To summon this rare you have to

1. 
   
2. Wait for Muck It Up world quest
3. 
   
4. Collect [item=177880] from mobs in the area
5. 
   
6. Use [item=177880] and each spot has a chance to summon the rare
   My Rare spawned when i used an item at 37.26 30.10 2 times in the same spot (showing it n the video)
   But then another character couldn't summon it at all for quite a while, so i suppose it's just random
7. 
   

Join a raid so you don't progress the quest because you can't get [item=177880] after the quest is done.
Drops [item=180588]

Rare is a part of Adventurer of Revendreth achievement.

/way 37.26 30.10 Bog Beast

Bog Beast video: https://youtu.be/NwhTUPHyODk

To farm the [item=177880], you cannot be in a raid group, but you can farm up to 20 of these on your own. After you have thrown four, you cannot throw anymore, so before you start throwing them, join a raid group so that it will not progress you on the quest. Throw what you have until the Bog Beast spawns. If you run out of Muck before it spaws, farm a new bunch and repeat.

[EDIT] It appears that you cannot start the raid group yourself, but if you join an already existing one, for instance via the LFG tool, you should be good.