Currently there are 6 Epic bosses in the Story mode. Provided below are the battle paths you must complete to face each Epic boss. Please keep in mind that you must be playing on Heroic difficulty and you must follow the path exactly.
Green Goblin[]
1st battle: Hydra *** LVL31
2nd battle: Mini Boss Hydra Power Armor *** LVL31
3rd battle: A.I.M. *** LVL 31
4th battle: Mini Boss Rhino **** LVL32
5th battle: Hydra **** LVL32
6th battle: Epic Boss Green Goblin **** LVL32
Graviton[]
1st battle: Sinister Syndicate *** LVL31
- Right to
2nd battle: Sinister Syndicate *** LVL31
3rd battle: Mini Boss Doctor Octopus *** LVL31
4th battle: Sinister Syndicate *** LVL31
- Right to
5th battle: Sinister Syndicate **** LVL32
6th battle: Epic Boss Graviton **** LVL33
Baron Strucker[]
Chapter 2 - A.I.M. For the Stars
1st battle: A.I.M. *** LVL33
- Left to
2nd battle: Mini Boss Tactical Force **** LVL34
3rd battle: Hydra **** LVL34
- Left to
4th battle: A.I.M. **** LVL34
5th battle: Mini Boss Hydra Four **** LVL34
6th battle: Epic Boss Baron Strucker **** LVL33
Ultron[]
Chapter 2 - A.I.M. For the Stars
1st battle: Mini Boss Crimson Dynamo *** LVL33
- Left to
2nd battle: Army of Ultron **** LVL34
3rd battle: Mini Boss A.I.M. Mecho-Trooper **** LVL34
- Left to
4th battle: Mini Boss Scientist Supreme **** LVL34
5th battle: Army of Ultron **** LVL34
6th battle: Epic Boss Ultron **** LVL33
Magus[]
Chapter 3 - Middle of Knowhere
1st battle: Minions of Thanos *** LVL35
2nd battle: Church of Truth *** LVL35
- Right to
3rd battle: Mini Boss Wraith **** LVL36
- Right to
4th battle: Ravagers **** LVL36
- Right to
5th battle: Church of Truth **** LVL36
6th battle: Epic Boss Magus **** LVL34
Nebula (Villain)[]
Chapter 3 - Middle of Knowhere
1st battle: Hydra *** LVL35
- Right to
2nd battle: Mini Boss Kree Sentry **** LVL36
3rd battle: Hydra **** LVL36
- Right to
4th battle: Mini Boss Tactical Force LVL36
5th battle: Minions of Thanos **** LVL36
6th battle: Epic Boss Nebula (Villain) **** LVL34