How would you optimize a multiclass druid/monk for DPR assuming Natural Weapons are considered Unarmed Strikes?

Question

Assuming that the Unarmored Defense, and Flurry of Blows features both work with the Druid's Wild Shape feature, what is the optimal DPR build for a Druid/Monk multiclass, and what are the notable break points for this particular multiclass?

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
PHB, page 67

Also assume that Natural Weapons are considered Unarmed Strikes, as per the the description of Alter Self:

Natural Weapons. You grow claws, fangs, spines, horns, or a different natural weapon of your choice. Your unarmed strikes deal 1d6 bludgeoning, piercing, or slashing damage, as appropriate to the natural weapon you chose, and you are proficient with your unarmed strikes.
PHB, page 212

Given access to any appropriate Beast, as well as having the option to choose either:
a) Multi-attack from the Beast entry, or
b) Extra Attack from the appropriate Monk level
What is the highest DPR possible?

Other assumptions:

• 27 Point Buy
• Official D&D Content only, excluding Unearthed Arcana

Answer 1

With your houserule, the DPR is huge

Natural Weapons are not Unarmed Strikes, making them so is a very unofficial, very unbalanced houserule.

Level selection

The best damage available in beast form is 24(4d8+6, Triceratops Gore) at +9 to hit, but you need 15 levels of Druid for it. To get the Extra Attack, you have to have 5 levels of Monk.
This is all of your 20 levels.

Starting Ability scores

for Human Variant with the Resilient(Con) for Concentration:
Str: 8
Dex: 16
Con: 14
Int: 10
Wis: 16
Cha: 10

ASI Monk 4: Sentinel, for out of turn DPR
ASI Druid 4: +2 Wis, for AC
ASI Druid 8: +2 Dex, for Flurry of Blows
ASI Druid 12: +2 Dex

Calculation

Assume AC 20 for the enemy, it is not unreasonable on level 20.

Official DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+11 to hit, 2 x (3.5+5) (Flurry of Blows) = 0.6 x 2 x 3.5 = 10.2
Alltoghether it is 34.2 vs AC20.
This is a very sudden jump however, at level 19 you either do not have Triceratops, or Extra Attack. Without Extra Attack you do 12 + 10.2 DPR, below a 17th level Monk.

Houserule DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+9 to hit, 2 x 24 (Gore as FoB) = 0.5 x 2 x 24 = 24
Alltoghether it is 48 vs AC20.
Extra Attack makes a much smaller difference now, as the FoB does the same damage as attacks done with your action. This is very unbalanced.

Compare it to 17 levels of Monk:
+11 to hit, 2 x (1d10+5) (basic attack) = 0.6 x 2 x 10.5 = 12.6
+11 to hit, 2 x (1d10+5) (Flurry of Blows) = 0.6 x 2 x 10.5 = 12.6
This all remains the same in the next 3 levels; 25.2 vs AC20.

Answer 2

Ability Scores

Operating on the assumption that you'll always be in combat using your Wild Shape feature, you can dump the three physical stats, hitting the minimum 13 for multi-classing as a Monk. I'll be using Half-Elf:

Strength: 8
Dexterity: 13(+1 from Racial Bonus)
Constitution: 8
Intelligence: 14
Wisdom: 16(+1 from Racial Bonus)
Charisma: 16(+2 from Racial Bonus)

At minimum you'll be awarded 4 Attribute Score Increases(ASI) over the course of the game, which should probably be attributed to Wisdom first. After that, assign feats or ASI as suits your interests.

Break Points

I did some of the calculations for average DPR, not accounting for to-hit. Assuming you house rule the natural weapons as unarmed strikes, the break points I've found are:

• Druid 2 (Wild Shape: Dire Wolf)
• Druid 6 (Wild Shape: Giant Elk)
• Druid 9 (Wild Shape: Ankylosaurus)
• Druid 15 (Wild Shape: Triceratops)
• Monk 2 (Ki)
• Monk 5 (Extra Attack)

Additionally add the following breakpoints without the aforementioned house rule:

• Monk 11 (Unarmed Attack D8)
• Monk 17 (Unarmed Attack D10)

Dire Wolf

Druid Level 2
Bite: 2d6 + 3(10)

Monk L2: 1d4 + 3 * 2(11)
Total Average: (Bite: 10) + (Flurry of Blows: 11) = 21
UA NW(Unarmed Natural Weapon) Total: 30

Monk L5: 1d6 + 3 * 2 + (Extra Attack: 2d6 + 3) Total Average: 10 + 13 + 10 = 30
UA NW: 40

Monk L11: 1d8 + 3 * 2 +(2d6 + 3)
Total Average: 10 + 15 + 10 = 35 UA NW: 40

Monk L17: 1d10 + 3 * 2 +(2d6 + 3) + Quivering Palm
Total Average: 10 + 17 + 10 = 37 AND Quivering Palm Instant Kill
UA NW: 40 + Quivering Palm

(Misses 1 ASI/feat)

Giant Elk

Druid Level 6
Ram: 2d6 + 4 OR
Hooves: 4d8 + 4 (Open Hand Technique combo*)
Monk L2: 1d4 + 4 * 2 Total Average: (Ram: 11) + (Flurry of Blows: 13) 24 OR Hooves(22)=35
UA NW: 35 OR 66(If not already prone: 55)

Monk L5: 1d6 + 4 * 2 + (2d6 + 4)
Total Average: 11 + 13 + 11 = 35 OR Hooves(22)=57
UA NW: 44 OR 88(77)

Monk L11: 1d8 + 4 * 2 + (2d6 + 4)
Total Average: 11 + 17 + 11 = 39 OR Hooves(22)=61 UA NW: 44 OR 88(77)

*Assuming your first strike hits, you can spend a ki-point to knock your target flat, or using your multi-attack, first strike with the Charge feature of the Giant Elk, then Hooves.

Giant Constrictor Snake

Druid Level 6 2d8 + 4 (grappled)
Monk L2: 1d4 + 4 * 2
Total Average: (13) + (Flurry of Blows: 13) = 26
UA NW: 39

Monk L5: 1d6 + 4 * 2 + (2d8 + 4)
Total Average: 13 + 13 + 13 = 39
UA NW: 52

Monk L11: 1d8 + 4 * 2 + (2d8 + 4)
Total Average: 13 + 17 + 13 = 44
UA NW: 52

Ideally go Monk12/Druid8 to max ASI/Feats.

Ankylosaurus

Druid Level 9
Tail: 4d6 + 4
Monk L2: 1d4 + 4 * 2
Total Average: (Tail: 18) + (Flurry of Blows: 13) = 31
UA NW: 54

Monk L5: 1d6 + 4 * 2 + (4d6 + 4)
Total Average: 18 + 13 + 18 = 49
UA NW: 72
Monk L11: 1d8 + 4 * 2 + (4d6 + 4)
Total Average: 18 + 17 + 18 = 54
UA NW: 72

(Misses 1 ASI/feat)

Triceratops

Druid Level 15
Gore: 4d8 + 6
Monk L2: 1d4 + 6 * 2 Total Average: (24) + (Flurry of Blows: 17) = 41
UA NW: 72
Monk L5: 1d6 + 6 * 2 + (2d8 + 4)
Total Average: 24 + 17 + 24 = 65
UA NW: 96

If Stomp instead of Flurry of Blows:
Stomp: 3d10 + 6
Monk L2: Total Average: (24) + (Stomp: 22.5) = 46.5
Monk L5: Total Average: 24 + 22.5 + 24 = 60.5
(Misses 1 ASI/feat)

vs

Given the numbers above, the most impressive DPR builds are Druid8/Monk12, and Druid15/Monk5, with the latter being slightly stronger.

However, this determines the average damage per round, which, while broadly useful, stems from a very limited resource of ki-points. Accounting for this, we see the average damage per round over time:

Here we see that, over time, certain builds maintain their DPR much better. Assuming you're averaging 6 rounds of combat each encounter, and 6 encounters per day, with one short rest, the Druid15/Monk5 pulls slightly ahead. This build works at full efficacy for ~1 encounter per short rest, with a reasonable performance on all other encounters. The alternative build of Druid8/Monk12 performs at full efficacy for ~2 encounters per short rest, so your best bet is to choose the build that best reflects the number of encounters you expect to have throughout a given day.

Conclusion

House Ruling the Natural Weapon as an Unarmed Strike still performs lower than the Sharpshooter Fighter builds available, while also being in melee range. You also give up one of the best capstones in the game, for a moderate increase in burst damage.

The clear front-runners for this multiclass are Druid8/Monk12 and Druid15/Monk5, for the priors access to one additional ASI, and the latter for its access to the Monk's Extra Attack feature. Realistically, if you plan to level this character up from 1, you'll probably be most successful with the Druid8/Monk12 as the other build doesn't come online until very late in your character's path.

Given that the levels of Druid are mostly front-loaded skills, you're most likely best off to take at least your first 2 levels in that, and assessing your needs as you level.

Deciding on which of these builds to choose largely depends on a number of factors:

• The number of encounters expected per day
• The number of short rests expected per day
• The likelihood of your party knocking someone prone
• The utility of druid spell slots
• The utility of Ki points (Stunning Strike and Open Hand Technique)
• How highly you evaluate the potential AC improvement with Unarmored Defense

Edit

As a point of reference, a Fighter 20 Battle Master using the Great Weapon Fighting feat averages 93.33 DPR exhausting no resources, while also having great utility, and impressive burst potential that replenish on the same short rest as the builds above. Druid 20 as a Mammoth has a DPR of 25 using Gore, and Monk 20 has a DPR of 22, or 66 within the first 20 rounds of combat a day.

Level 20 Fighter GWF Battle Master
4 * ((2d6(reroll 1/2) + 5) + 10)
4* ((8.33 + 5) + 10)
4* (23.33)
DPR: 93.33
First Round
93.33 + Second Wind(23.33) + Superiority(6.5)*4
142.66
Second Round
116.66

Answer 3

The best DPR would possibly be for moon Druid 4, monk 16, wood elf. 8 13 9 13 16 15 4 ASIs give us 8 13 9 13 20 15

At this point, there are 2 feats remaining. I choose mobile and wood elf magic My speed is 35 + 25 + 10 + 10 = 80. This is relevant. Cast longstrider before combat. In the first round of combat, cast enhance ability and wild shape into a toad. Second round: bite attack (twice) +8 to hit and 12 damage. Against a foe with ac 20, chance of hitting once is 84%, 14.4 average damage so far. Bonus action to step of the wind dodge. The speed is now 140. Standing jump 14 times. Enhance ability makes you immune to fall damage, but not the grappled target. 16d6 averages to 56 damage. All subsequent rounds: 32d6 unavoidable damage is 112 per round.

Answer 4

Interesting note: The 14 Monk/ 6 Druid, circle of the land grassland has some ridiculous mobility options, though would be comparatively lacking in DPR. The combination of monk speed buffs, land strides removal of difficult non-magical terrain, and the ability to cast haste on yourself to double your speed and grant an addition action (which you can use to dash or disengage as well as attack) could make you an untouchable whirlwind on the battle field.

55 base speed, x2 from haste, with action (dash) + hasted action (dash) + Bonus Action using a ki point to (dash) + movement= 440 ft per round. Take the mobile feet and this speed increases to 520, cast longstride on yourself before combat and it increases to 600, and if your DM is so kind as to give you Boots of Speed, you could move at 1200 ft per round. Finally, if youre a Tabaxi, you can use Feline agility to double your base speed for a turn, moving at a whopping 2400 feet a round, 400 ft/sec, or 272 mph. And you could do all of this except Feline Agility every round for a minute.

To put this into perspective, if you were to simply run for 3 min using all of your casts of haste, without using feline agility due to its movement restriction, you could run about 7 miles during that time, completely ignoring terrain. Considering a day's travel is considered to be 30 miles...you get the picture.

Edit: Oh, and if you were to wild shape into a creature with a faster move speed, say, a horse which has a move speed of 60 ft, this would also apply. According to the wild shape, "You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so." Therefore the mobility feat, extra monk speed, Landstrides ignorance of terrain, longstride and haste, as well as Feline agility, could be applied. Also, because "You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. [pg. 67 - PHB], " your horse can wear your boots of speed. The point is that this can get very ridiculous very fast.