Neverwinter

Note: This isn't a guide on how to beat Demogorgon, or a strategy as such. This is to point out the key differences between the regular Demogorgon and the Epic version, which will be useful to know before going into the fight.

To unlock Epic Demogorgon (EDG), you must complete the "Process Faerzress" task 10 times in the Underdark Campaign window. Once you've unlocked EDG, you can queue for it anytime, anyplace. You do not need to be in Mantol-Derith, or wait until X:30 on the hour. The minimum Item Level for EDG is 2500, but I personally recommend at least 3000 for the whole group, as otherwise Gold may be out of reach. I'm not going to explain how to raise your IL, as there are many guides on that already (including the one literally right above this post by The Nerevarine).

Most of the fight is relatively the same as the regular Demogorgon, but I will list the key differences:

Enemies are now Epic Lv72. That's 2 levels higher than you will be, so they will do more damage and take less damage overall. They are weaker than Epic enemies in Icewind Dale or Well Of Dragons though, so that's a bonus.

A new enemy, Paranoid Delusion, can spawn at any time in any place, usually in pairs. They do not appear from rifts though. These are only lesser enemies with 1 health section, so they're easy to take down. They don't hit hard, so you should ignore them and let AOE DPS deal with them as collateral.

There are less Sanity Wells now. The first stage only has 1 Sanity Well, at the start of the area where you teleport in. The second stage will have 2; one at the start, and another randomly to the left or right of the start. The final stage will have 3; two at the start and a third at the back of the room near Demogorgon.
Because there are so few compared to the regular fight, everyone should make sure to keep enemies out of the Sanity Zones, as they don't work while enemies are within (they will turn red instead of blue). Many classes need their Encounter or Daily powers to support the group or DPS, and when they can't clear their Madness to use them, the whole group suffers.
EDIT: It seems that more Sanity Wells will appear temporarily during Stage 2 while Demogorgon is using his Madness beam attack. A message will pop up informing you to seek shelter, and there will be more Wells around the room.

You will constantly be taking damage while in the arena. I personally take between 12K and 15K damage every 4 seconds, and I have 5800 Defense (TR) with the 1 piece set bonus of Drowcraft (Demons deal 10% less damage). The damage you take seems to be related to the amount of Madness you have, as with no madness I only take around 6K damage. You should try to get a group with a good healer or a bubble or two to prevent this damage, so you can focus more on DPS with less deaths.

The first stage is mostly the same. The Paranoid Delusion enemies do not fill the completion bar, so never concentrate your attacks on one. They don't hit hard, so you can usually ignore them. Try to only close epic (huge purple ones) or rare (medium blue) rifts. Ignore the smaller green rifts, as the small reward they give is not worth the time it takes to kill the enemies relative to purple or blue. Remember to lure all enemies to a communal area where the DPS classes are. It'll greatly increase damage due to Combat Advantage with everyone together, and also let support classes easily buff and debuff everyone at once.

The second stage of the fight, Goristro, is more difficult. There is now only 1 golden portal on the walls instead of 2. If you do not have enough DPS as a group to kill Goristro without using the portals, make sure you know where the golden one is at all times. If you miss and hit one of the 3 black portals, you will need to lure him into a golden one 4 more times to offset the healing Goristro receives from a black portal. You can ignore the Paranoid Delusions. Keep in mind that once Goristro has taken a bit of damage, stalactites will begin to fall from the ceiling (similar to Lostmauth). There will be tiny red circles on the ground indicating they are going to fall, and they hurt, so try to avoid them.
NOTE: Apparently this stage of the fight can glitch and be skipped. If it's skipped, it will be like you didn't get any medal on it from what I've heard, and make Gold impossible to achieve. I'm not sure what causes it (it's never happened in the 10+ runs I've done), but some people say it happens if you attempt to enter the arena before the boss has spawned in and you see him. It's best to wait at the blue barrier until you can clearly see Goristro in the arena.

The third stage is still the same; simply kill Demogorgon. Remember to equip and use your Lanterns whenever you can for the damage boost.

TIPS:
If you can get the Drowcraft Armor from the trader in Mantol-Derith, the set bonus from that will greatly help in this fight.

You will get Twisted Ichor in the bonus chest at the end if you achieve Silver or above. 1 Ichor with Silver, and 2 with Gold. You will need to use a Greater Demonic Key or a Legendary Dragon Key to obtain this Ichor. It's used to buy the Twisted Artifact Weapons from the trader in Mantol-Derith.
As a final note, if you get Gold in EDG the Twisted Artifact weapons have a small chance to drop. If this is a set you want to get, do NOT buy one of the weapons using Ichor until you have enough to get both weapons at once. Imagine if you had just bought one of them, and then got a Gold rank and had the same weapon drop as a reward. You would be pretty annoyed.


For the basics of the Demogorgon fight in general and what to do, you should check out The Nerevarine's solution here: Solution for Tamer of Demogorgon In Neverwinter

Master of Demogorgon
You earned a Gold victory in Epic Demogorgon

To unlock Epic Demogorgon

Use the Campaigns menu, you can "Process Faerzress" once every 18 hours using the currency of the same name that you earn from completing skirmishes, raids, and encounters in the new Underdark content. Running the task ten times will enable access to the "Unlock Epic Demorgorgon" task. You can queue for Epic Demogorgon once the task is unlocked from anywhere at any time, however you'll need a 2,500 item level. After that you must have 2,500 item level!

Instead of me writing a very long guide, here is a video I made that will show you what you need to do to unlock this. The video is in 1080p 60fps with commentary explaining what you need to do at each step. I hope this guide helps!