Note: I'm using the new rules for Bugbear from Mordenkainen Presents: Monsters of the Multiverse. Primarily, this removes the once/turn resstriction.

Bugbear Champion gets ~189 damage to one target

I lot of this is in anydice and I'm going to need someone to triple check me; thanks to @Someone_Evil for the assist on the Great Weapon Fighting Dice.

The Character (Decisions)

• Bugbear Fighter - Str 17, Dex 13, Con 14, Int 10, Wis 12, Cha 10
• Level 1 Great Weapon Fighting style
• Level 3 Champion Fighter
• Level 4 +2 Str - Str 19
• Level 6 +1 Str, +1 Dex - Str 20, Dex 14
• Level 8 Mounted Combatant
• Level 12 Great Weapon Master
• Level 14 Lucky
• Level 16 Whatever, more con or dex?

The Idea

Alright, I will admit that I couldn't reliably figure out the exact chance that you get zero crits and, thus, no 7th attack from Great Weapon Master, but I think it's around 10%, so I'll apply the entire total to a 10%/7 dummy-tax (about 2%) unless someone else can figure it out.

Mounted Combatant gives us advantage on every attack, and since we're going first in the initiative, we get an extra 2d6 from Bugbear's Surprise Attack. Those dice are added to the weapon damage, so they should benefit from Great Weapon Fighting Style, but there's an argument to be made that they don't. Use the bonus damage from Great Weapon Master on every attack.

The Calculations

I couldn't write a reliable program to trigger the Lucky feat on a failed roll, so I've configured the anydice program to simulate Lucky by doing three rolls at 'best of 3d20'.

Basically 7 attacks with a 25% chance to do 4d6+15, and a 15% chance to do 8d6+15 (the d6 all, of course, actually being GWF d6, 4.167 damage instead of 3.5).

That should bring us to 193.34 expected damage, and then the aforementioned dummy-tax brings us down to 189.47.

Feel free to take a look at the AnyDice.

Some other things to note

• Hex is in another answer, but in my testing brought it slightly lower than getting a 7th attack. (hex damage exceeds the bonus attack at level 20)
• It's worth noting level 20 (extra attack(3) bumps us over the Meteor Swarm benchmark - ~240 expected damage
• Max Damage on the meteor swarm is 240/target and max damage for fighter is 441

Thing that would improve this answer

• Get exact difference for the chance of not getting the 7th attack.
• Improve Lucky "logic" from just being 3 rolls at best of 3d20
• Find a good use for the last feat.
• Martial Adept - Precision Attack potentially adding another hit
• Savage Attacker, potentially increasing a poor damage roll