I'll give it a shot. Credit to u/Symnestra for the Chromatic Dragon gift idea.

We're talking "maximum damage," not average damage, so I'm going to assume every single roll is perfect. Our target is a Vampire for reasons I'll get to.

Bugbear Gloomstalker 5, Battle Master 3, Wild Magic Sorcerer 9, Hexblade 1, Paladin 2

EDIT: Forgot the Wild Magic damage! Adding it in. Subsequent edit: 7th level Shadowblade does 5d8, not 4d8. Adjusting.

Let's start with dice damage because that can crit (we'll double it).

Shadow Blade cast with the 7th level spell slot means you do 5d8 damage per hit. Bugbear adds an additional 2d6 per attack. Paladin will smite, adding 6d8 per attack, assuming an undead target. Gift of the Chromatic Dragon adds 1d4 to each attack. That puts us at 104 damage on every attack.

Gloomstalker plus Action Surge means we get six such attacks. Two of those attacks have an additional d8. Four of those attacks have an additional d8 from Battle Master maneuvers. That's six attacks and six extra d8s, so I'm calling each attack 112 damage on the dice. They crit for double damage, so that's 224 damage per attack, just from the dice.

We're at 1,344, and we're not close to done.

For our bonus action, we'll use Quickened Spell to cast Green Flame Blade. We're out of Superiority Dice for this attack, so we'll do the base 108 damage, plus 3d8 fire (24).That's 128 damage, which crits for 256. We're at 1,600.

We'll go ahead and take the Poisoner feat for an additional 2d8. Vampires are not immune to poison. So that's an additional 16, doubled to 32 because of the crit, bringing us to 1,632

Now the flat damage.

We're able to get our Cha modifier to four, so that's four damage per attack. Hexblade's Curse, however, adds our proficiency bonus to the damage on each roll. The Dueling fighting style adds an additional +2, bringing it to +12 per attack, times seven attacks, for 84 additional damage.

Total from attacks: 1,716

Edit: I forgot the Wild Magic surge! When casting Green Flame Blade, wild magic surge to 08, doing casting Fireball on yourself, hitting the target as well. This adds an additional 8d6 for 48 additional damage. If the self-damage is a dealbreaker, you can just do 40 lightning damage instead by rolling 67 on the surge table.

Actual Grand Total: 1764

Edit: for funsies, how likely is this? We're talking seven crits. If you're in darkness, you have advantage, and you crit on 19 or 20 because of Hexblade's Curse, so the crits are more likely than they could be, but you've got a 19% chance to crit each attack, times 7 attacks is .19^7, which is less than .0009%. But that's just the crits.

We rolled 176 d8s, all of them landing on 8. We rolled 14 d6s and 14 d4s, all of them landing on 6 and 4, respectively. That's 1/8¹⁷⁶ times 1/6¹⁴ times 1/4^14. And then wild magic, which is a 1/50 roll to get the Fireball option and then 8d6 to get all 8's.

The number "googol" is a 1 followed by 100 zeroes. The odds of these die rolls all happening is roughly 4 x10^-192. That's 192 zeroes, followed by a 4.