I'm trying to net (ha) as many thrown attacks using the ever-so-special net. There's a big hurdle, however.

[snip]... When you use an action, bonus action, or reaction to attack with a net, you can make only one attack regardless of the number of attacks you can normally make.

How many net attacks can be made by a character in one round?

Constraints:

• Point Buy (27 point) attribute generation


• No magic items


• Official sources only (No 3rd party; No UA; Eberron is okay)


• Any level


• Multiclassing is allowed


• Does not have to be good at net throwing, just able to throw a lot of them


• Feats are allowed.


• You may have one noncombat preparatory round


• Answer goes to whomever can throw the most nets in one round
  
  
  • ties go to whoever is better at throwing nets
  
  


• Help from another is not allowed

4 nets in one round in those conditions

Fighter 2 / War Cleric 1 with the Dual Wielder feat.

Round 1.

Use your action to ready an attack action with one of your nets.

Round 2.

Right before your turn, you use your reaction to throw your first net. Then on your turn you use an Attack action to throw your second net. You then use your free object interaction to draw two nets (instead of one because of the dual wielder feat). Using the the war priest feature of the cleric's war domain allows you to make an attack as a bonus action, allowing you to throw a third net.

War Priest. [...] When you use the Attack action, you can make one weapon attack as a bonus action. [...]

You then use Action Surge to make another attack and throw your fourth and last net.

Note

The Cleric's War Domain (or a Path of the Berserker Barbarian's frenzy) is needed, because as written, the dual fighting does not work with nets. They aren't melee weapons, they are considered ranged weapons.

The ready action can be replaced by an extra action given by the spell haste, but both cannot be used together simply because you run out of nets in your hands.

Extra

The absolute maximum of nets that you can throw in a single round seems to be 6, but for that you need allies.
Use the spell Shapechange to assume the form of a Marilith so that you can have 6 nets ready in your hand. Use the haste spell to add another action allowing you to throw a fifth net. Then another one of your allies uses the Commander's Strike maneuver, allowing you to use your reaction (that you gained back at the start of your turn) to throw a sixth and final net.

5 nets

Though it relies on quite a bit of luck.
Class: Sorcerer 5 (wild) / Rogue 3 (thief) / Fighter 2, Dual Wielder feat.

Preparatory Round:

Cast Quickened Haste as a bonus action, and roll a 01-02 on the wild magic table (roll at start of every turn for next minute). Draw two nets. Ready an action to throw a net.

Execution Round:

Reaction triggers, throw a net. (1)

Execution Round, during turn:

Roll 81-82 on wild magic table (take one additional action immediately)

Wild Action - throw a net. Draw two nets. (2)

Regular Action - throw a net. (3)

Haste Action - throw a net. (4)

Bonus Action - Cunning Action - Draw two nets.

Action Surge - throw a net. (5)

The most a single character can throw is 38

The Character

The character, let's call her Celeste from here on out, that can throw the most nets is a level 20 character with the following classes:

• Sorcerer (Wild Magic) 17 ------ (for Wild Magic Surge, Flexible Casting, Quickened Spell, the Dual Wielder feat, and spells)


• Fighter 2 ------------------------ (for Action Surge)


• Cleric (War Domain) 1 --------- (for War Priest)

..., at least 20 Wisdom, and the Boon of High Magic

The Preparation

For Celeste's preparation round, she follows the following procedure:

1. Cast haste with the Quickened Spell Metamagic.


2. Roll a 1 on the d20 for Wild Magic Surge.


3. Roll 01-02 on the d100 for Wild Magic Surge.


4. Take the Ready action to throw a net at the start of the round.


5. Use the effect of haste take the Use an Object action and draw your first two nets of many.

The Throwing

Now that Celeste has prepared, she throws her nets:

• First off, she throw the net she had readied, then her turn begins:

Normal Turn

1. Roll 81-82 on the d100 for Wild Magic Surge.


2. Throw a net. ---------------------------------------------------------------------- [total so far: 2]


3. Cast ray of sickness using Bend Luck on the attack roll.


4. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.


5. Throw a net. ---------------------------------------------------------------------- [3]


6. Draw two nets as part of your movement.


7. Use Action Surge.


8. Cast time stop rolling a 4 on the 1d4.

Turn One of Time Stop

9. Repeat steps 1-6, but use Quickened Spell Metamagic on ray of sickness and target an empty space¹. ---------------------------------------------------------------------------- [5]


10. Throw a net at an empty space. ------------------------------------------------ [6]


11. Use the haste action to throw a net at an empty space. --------------------- [7]

Turn Two of Time Stop

12. Roll 81-82 on the d100 for Wild Magic Surge.


13. Take the Use an Object action to draw two nets.


14. Throw a net at an empty space. ------------------------------------------------ [8]


15. Use War Priest to throw a net at an empty space. ---------------------------- [9]


16. Draw two nets as part of your movement.


17. Use the haste action to throw a net at an empty space. --------------------- [10]

Turn Three of Time Stop

18. Roll 81-82 on the d100 for Wild Magic Surge.


19. Throw a net at an empty space. ------------------------------------------------ [11]


20. Cast ray of sickness with the Quickened Spell Metamagic on an empty space (use Bend Luck).


21. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.


22. Take the Use an Object action to draw two nets.


23. Throw a net at an empty space. ------------------------------------------------ [12]


24. Use the haste action to throw a net at an empty space. --------------------- [13]


25. Draw two nets as part of your movement.

Turn Four of Time Stop

26. Repeat steps 9-11. --------------------------------------------------------------- [17]

Turn Five of First Time Stop

27. Roll 81-82 on the d100 for Wild Magic Surge.


28. Take the Use an Object action to draw two nets.


29. Use the haste action to throw a net at an empty space. --------------------- [18]


30. Use War Priest to throw a net at an empty space. --------------------------- [19]


31. Draw two nets as part of your movement.


32. Repeat steps 8-20. -------------------------------------------------------------- [33]

Turn Five of Second Time Stop

33. Roll 81-82 on the d100 for Wild Magic Surge.


34. Take the Use an Object action to draw two nets.


35. Throw a net at an empty space. ------------------------------------------------ [34]


36. Use War Priest to throw a net at an empty space. --------------------------- [35]


37. Draw two nets as part of your movement.


38. Use haste action to throw a net. ----------------------------------------------- [36]

Normal Turn

39. Use haste action to throw a net. ----------------------------------------------- [37]


40. Use War Priest to throw a net. ------------------------------------------------- [38]

The Probability

First of all, your GM has to allow you to roll on the Wild Magic Surge table (to recover Bend Luck) every time you cast ray of sickness, and to have the chance when you cast haste. This is already extremely unlikely. Then there are the actual rolls.

I changed the procedure a few times to try to get the best odds of this completing. The rolls necessary are listed below:

• 1 on the d20 one time
  
  
  • (1/20)
  
  


• 1 or 2 on the d100 one time
  
  
  • (1/50)
  
  


• 81 or 82 on the d100 nineteen times
  
  
  • (1/50)¹⁹
  
  


• 4 on the d4 two times
  
  
  • (1/4)²
  
  

Multiplying these probablities comes to one chance in about three undecillion (3,051,757,812,500,000,000,000,000,000,000,000,000 to be exact)

¹ You have to target an empty space with ray of sickness and the nets during time stop so that the spell doesn't end:

This spell ends if one of the actions you use during this period, or any effects that you create during this period, affects a creature other than you or an object being worn or carried by someone other than you.

Attacking empty spaces is perfectly valid according to the Making an Attack section of the Basic Rules (emphasis mine):

1. Choose a target. Pick a target within your attack's range: a creature, an object, or a location.