Multiclass DPR

[Giant2005]

By now most people probably have a good grasp of how the pure classes compare to one another when it comes to damage, but multiclassing has been much too varied and much too customizable for fair comparisons to be made on anything but a personal scale. I thought I'd post the results of a few combinations here, so you can get a sense of how powerful muticlassing is (or isn't in some cases) with respect to pure damage.
This is just theory - I wouldn't recommend trying to play many of these combinations, as although they are effective at level 20, their effectiveness on the way to 20 would be much more in question.

In order to set a baseline, I'll first post the DPR values of two pure classed builds - the king of DPR: The Barbarian that goes balls to the wall with such grand enthusiasm that the damage inflicted to the enemy is only matched by the suffering he inflicts upon himself; and a more standard character: A pure Warlock.

*Note all damage is calculated against a single enemy of AC 18. That enemy has +4 to all saves when relevant.

20 Barbarian (Berserker): 73.4 DPR with 24 Str, a Greataxe; using Rage, Frenzy, Reckless Attack and Great Weapon Master.

20 Warlock (Any): 41 DPR with 20 Cha, Eldritch Blast, Agonizing Blast, and Hex.

And now, for the multiclass builds (I worked out the DPR for every combination that I could think of which had a chance of increasing DPR by exploiting multiclass combinations - if you can think of any other possibilities that I missed, go ahead and mention it and I'll add them to the list):

2 Warlock, 7 Fighter (Eldritch Knight), 11 Rogue (Any): 57.025 DPR with 20 Dex, 20 Cha, using Eldritch Blast and a Heavy Crossbow for bonus action. Increases to 73.18875 DPR if Darkness has been precast (with Devil's Sight Invocation).

2 Warlock, 18 Sorcerer (Any): 61 DPR using Quickened EB + EB. Increases to 80.73 if Darkness has been precast. Increases to 82 if Hex has been precast.

11 Ranger (Beast Master): 9 Rogue (Thief): 62.375 DPR with 20 Dex, using Flying Snake, Heavy Crossbow (and qualifying for a Sneak Attack), Flying Snake's venom applied with Fast Hands. Increases to 70.775 DPR if Beast Bond has been precast.

16 Cleric (Any with Divine Strike), 3 Rogue (Arcane Trickster), 1 Fighter: 37.275 DPR with 20 Str, 20 Wis, a Rapier, Magic Initiate (Booming Blade), and a 2nd level Spiritual Weapon. Increases to 50.775 DPR if the target moves. Increases to 106.125 DPR if a 9th level Spirit Guardians has been precast and is double-tapped. Increases to 119.625 DPR if the target moves, and a 9th level Spirit Guardians has been precast and is double-tapped.

12 Paladin (Oathbreaker), 4 Fighter (Champion), 3 Barbarian (Berserker) 1 level of anything: 80.2 DPR with 20 Str, 20 Cha, and a Greatsword; using GWF, Rage, Frenzy, Reckless Attack, and GWM.

12 Fighter (Eldritch Knight), 6 Bard (Lore - Magical Secrets: Find Steed, Elemental Weapon), 2 levels of anything: 59.261 DPR with 20 Str, GWF, GWM, Polearm Mastery, Mounted Combatant (and a mount larger than your target) and a Halberd. Increases to 88.6 DPR if a 5th level Elemental Weapon has been precast.

10 Bard (Valor - Magical Secrets: Swift Quiver), 1 Fighter, 9 Rogue (Any): 46.96 DPR with 20 Dex, Archery Fighting Style, Sharpshooter, Crossbow Mastery and a Hand Crossbow. Increases to 62.98 DPR if Swift Quiver has been precast and you use a Heavy Crossbow.

9 Sorcerer (Draconic), 11 Rogue (Arcane Trickster): 65.881 DPR with 20 Dex, 20 Cha, Green Flame Blade, Quickened Hold Monster (Can only quicken 6 Hold Monsters per long rest), and a Rapier.

12 Paladin (Crown), 8 Sorcerer (Draconic): 53.5 DPR with 20 Str, 20 Cha, Green Flame Blade, Quickened Green Flame Blade, GWF, and a Greatsword. Increases to 107.05 DPR if a 7th level Spirit Guardians has been precast and is double-tapped.

9 Rogue (Arcane Trickster), 6 Sorcerer (Draconic), 2 Warlock (Any*) 3 Fighter (Champion): 60.3 DPR with 20 Dex, 18 Cha, Dueling, Green Flame Blade, Quickened Green Flame Blade, and a Rapier. Increases to 77.61 DPR if Darkness has been precast (with Devil's Sight Invocation).
*I wasn't intending on including any homebrew options, but if you use the UA's Undying Light patron, the DPR increases to 65.9 without Darkness and 84.89 with Darkness.

[bid]

I'm not sure how you consider limited resources in your concept of DPR. For instance, one use of frenzy will last an entire combat while each quickened metamagic eats resource as a 2nd slot (and a fractional bonus action).

Do you have a rule of thumb on how much of your resouces you can spend on a single battle, and across how many rounds you must spread them?

[Gtdead]

Just a correction, sorcerer doesn't get access to foresight, so 2 warlock /18 sorcerer doesn't get it.

[Kryx]

Level 20 DPR numbers without showing the math really lacks value imo. Without seeing full numbers it cannot be corrected (I know from experience that there are lots of little errors which make a big difference).
In my DPR of Classes if you only look at 20 many classes look very different than they do at 17, 11, or 5.

Also of note is Bid's question of what metrics you use - I would recommend using ones similar to the DMG based numbers I came to (https://docs.google.com/spreadsheets/d/1d-9xDdath8kX_v7Rpts9JFIJwIG3X0-dDUtfax14NT0/edit#gid=2091322934).

[Giant2005]

Just a correction, sorcerer doesn't get access to foresight, so 2 warlock /18 sorcerer doesn't get it.

Fair enough! I'll switch that out for Darkness - it doesn't change any of the calculations.

[Giant2005]

I'm not sure how you consider limited resources in your concept of DPR. For instance, one use of frenzy will last an entire combat while each quickened metamagic eats resource as a 2nd slot (and a fractional bonus action).

Do you have a rule of thumb on how much of your resouces you can spend on a single battle, and across how many rounds you must spread them?

All of the listed abilities were used (as well as permanent abilities like Improved Divine Smite that aren't optional once gained so I didn't list them).
Basically, I only chose classes with abilities that could potentially be used on every round of combat, over multiple combats in a day (and if some of the chosen classes have abilities that don't qualify on those grounds, I ignored them). 6 Quickened Hold Monsters may be pushing that limit a little bit as it could only amount to 6 rounds of combat which obviously does not qualify for multiple combats in a day, but it could also potentially last for 60 rounds of combat, so I included it anyway. It is the only combination that might not be able to maintain its DPR for the distance (those that use Frenzy would also suffer from overuse, but the option is there if they don't mind the consequences).

[Citan]

By now most people probably have a good grasp of how the pure classes compare to one another when it comes to damage, but multiclassing has been much too varied and much too customizable for fair comparisons to be made on anything but a personal scale. I thought I'd post the results of a few combinations here, so you can get a sense of how powerful muticlassing is (or isn't in some cases) with respect to pure damage.
This is just theory - I wouldn't recommend trying to play many of these combinations, as although they are effective at level 20, their effectiveness on the way to 20 would be much more in question.

In order to set a baseline, I'll first post the DPR values of two pure classed builds - the king of DPR: The Barbarian that goes balls to the wall with such grand enthusiasm that the damage inflicted to the enemy is only matched by the suffering he inflicts upon himself; and a more standard character: A pure Warlock.

*Note all damage is calculated against a single enemy of AC 18. That enemy has +4 to all saves when relevant.

20 Barbarian (Berserker): 73.4 DPR with 24 Str, a Greataxe; using Rage, Frenzy, Reckless Attack and Great Weapon Master.

20 Warlock (Any): 41 DPR with 20 Cha, Eldritch Blast, Agonizing Blast, and Hex.

And now, for the multiclass builds (I worked out the DPR for every combination that I could think of which had a chance of increasing DPR by exploiting multiclass combinations - if you can think of any other possibilities that I missed, go ahead and mention it and I'll add them to the list):

Hi! Interesting thread. :)

You already put out my rocking Flame combination (Sorcerer + Warlock + Rogue) and Elemental combination (Elemental Weapon) but I think there are some nuances to add.

Ranger Hunter 11 with Volley + Paladin Devotion 3 + Bard 6 could become a beast for one fight: cast Sacred Flame and lvl 5 Elemental Weapon on your bow and you're out for serious damage. Too bad it's impossible to make any decent guess about actual damage since the number of enemies in potential area is too variable. On a crowd it would be deadly though. :)

In the case a DM overrun the ruling about "CHA bonus applies once only", two powerful builds would be...
Draconic Sorcerer 16 / Undying Light Warlock 2 / Fighter 2, casting 2*lvl 6 Scorching Ray and additional Eldricht Blast as quicken. But this is only for nova damage.
Or Paladin 12 + Undying Light Warlock 1 + Sorcerer 7: cast Elemental Weapon on your GWM and stack 5+5+10 on each hit in addition to usual Paladin damage and potential smite ability.

Monks benefit heavily from Ranger and Rogue dips, the first for potential extra attack that becomes easier to land at higher level for any Monk (or very easily for Sun Soul), the second for allowing several useful bonus action without ki cost.

Beyond that, I think you covered most areas (although I don't have time to think about every possibility for now ;)) at least for sustained damage. Nice work!