Highest damage builds at tiers 1 (1-5) and 2 (6-10).

So far I have this: (if you go first, so pump that Dexterity and take the Alert feat if you can)

Sorcerer 3 / Warlock 2

Bugbear

1) Pre-cast Hex (This is the tricky part)
2) 2nd level Magic Missile (Quickened)
3) Eldritch Blast (with Agonising Blast, of course)

Total of (1d4+1 (Magic Missile) + 2d6 (Surprise Attack) + 1d6 (Hex)) per dart.

Using a 2nd level spell slot we have 4 darts. 
= 4(1d4+1+2d6+1d6)

If we quicken this, we can use our action to Eldritch Blast.

Assuming a CHA score of 16,

(1d10 (Eldritch Blast) + 3 (Agonising Blast) + 2d6 (Surprise Attack) + 1d6 (Hex)) x 2 (Cantrips get a second beam at level 5)

Both of these together, we get an average damage score of:

4(1d4+1+2d6+1d6)+2(1d10+3+2d6+1d6)

or 94 points of damage.

At level 5.

The downsides are:

1) Pre-casting Hex might not always be an option. If it isn't, you can still deal 73 (4(1d4+1+2d6)+2(1d10+3+2d6)) damage.
2) You can only really do this twice or thrice per long rest.
3) It relies on you going first. If you don't go first, it drops to 31 without Hex (4(1d4+1)+2(1d10+3)) or 52 with it. Still respectable burst damage though.

When you run out of resources (which you will, inevitably), you still have the trusty Eldritch Blast + Hex (if you have a spell slot or two) combo to keep you going. 

Cheers!

Magic missile doesn’t make an attack roll, so it won’t trigger the bugbear’s sneak attack ability. 

Oh, sorry! Thank you for that!

How about this one:

Bugbear

Sorcerer 5

1) Quicken a 3rd level Scorching Ray
2) Use your action for Firebolt

Total of (2d6 (Scorching Ray) + 2d6 (Surprise Attack) per ray.

Using a 3rd level spell slot, we get 4 rays. With the Quickened Firebolt, we get:

4(2d6+2d6) + (2d10+2d6)

or around 74 damage.

The only real question here is if you would cast this over Fireball.

It's a pity that Sorcerers don't get Hex or Spirit Shroud...

How about Hexblade Warlock-2 + Sorcerer-3 + Bugbear.

If you allow precasting:

Assuming CHA = 3
HBC = +3 damage on each attack
Hex = +1d6 damage on each attack
Bugbear = +2d6 damage on each attack
Action = EB = 2 rays of 1d10+3
Bonus Action = Quickened Scorching Ray = 3 rays of 2d6

= 2*(1d10+3) + 3*(2d6) + 5*(3+1d6+2d6) = 2d10+21d6 + 21 = 167 damage

Alternatively, Bugbear + Beast Barbarian-3 + Monk-2

Strength = +3
Pre-Rage
Bugbear = +2d6 damage on each attack
Rage = +2 damage on each attack
Action = Attack with Claws 2x
Bonus Action = Flurry of Blows

= 2x(1d6+3+2+2d6) + 2x(1d4+3+2+2d6) = 88 damage.

Much less than the above, but this one is much more viable to actually pull off since Raging can be done while Hidden without alerting enemies and only 1 pre-cast is necessary rather than the 2 for HexSorLock. It can also be used 3x per day assuming you pull off a SR. Plus since a lot more of this is flat modifiers your average or minimum damage is still pretty high.

For Tier 2 I'll submit:

Bugbear + Beast Barbarian-5 + Mercy Monk-3 + Fighter-2 (swapping one monk level for Hexblade is better but you're super MAD so wouldn't be at all viable for anything other than DPR, plus you'd need 3 set up rounds which is pretty ridiculous TBH)

Bugbear = +2d6 damage on each attack
Rage = +2 damage on each attack
Action = 3x attacks - 1x Longsword with Dueling + 2x Claws
Action Surge = 2x actions
Flurry of Blows = 2x unarmed strikes
Mercy Monk = +1d4+Wis once
Str = +3
Wis = +3

Longsword = 1d8+2+3+2+2d6 = 27
Claw = 1d6+3+2+2d6 = 23
Unarmed = 1d4+3+2+2d6 = 21

= 188+7 = 195 damage (minimum damage is still: 80 if all attacks hit)

RAW you cannot get the Dueling bonus. The claws count as a simple melee weapon. Instead of Dueling get Unarmed Fighting so your monk strikes hit harder.

Wouldn't you get more damage out of two swings with a greatsword?

Quote from Wysperra >>

RAW you cannot get the Dueling bonus. The claws count as a simple melee weapon. Instead of Dueling get Unarmed Fighting so your monk strikes hit harder.

Wouldn't you get more damage out of two swings with a greatsword?

 Oh yeah, forgot about that that would be a better option. Re: Claws no. Claws > Greatsword:

Greatsword = 2d6+STR+Rage x 2 = 7+3+2 x2 = 24
Claws = 1d6+STR+Rage x 3 = 3.5 +3 +2 x3 = 25.5

longsword 1d8 + claws 2d6 x2

greatsword 2d6 x4

Paladin 8 / Fighter 2

Greatsword 2d6 + Divine Smite +3d8  x3

Greatsword 2d6 + Divine Smite 2d8

Quote from Wysperra >>

longsword 1d8 + claws 2d6 x2

greatsword 2d6 x4

 

 You're missing out the Rage damage and the Strength modifier which longsword + claws get to add 3x, whereas greatsword only gets to add it 2x. Barbarians are great because they depend less on dice for damage than any other class, it's all about the modifiers. It's why PAM is incredibly popular among barbarians, getting that 3rd attack per turn is so so powerful when you're add 5 or 6 flat damage to that attack every time.