A character can pull off ∞ with perfect rolls (or as many nets as you have)
Graças a @Ko_sct in the comments for this strategy

O personagem

Any level 17 Wild Magic sorcerer can accomplish this using:

• Surto de Magia Selvagem
• parar o tempo
• the Dual Wielder feat

The Loop

1. Elenco parar o tempo
2. Trigger Wild Magic Surge
3. Roll a 1 on the d20
4. Roll a 59-60 on the d100
5. Throw nets throughout parar o tempo until the last turn of parar o tempo
6. Draw two nets (possible because of Dual Wielder) as part of your movement when necessary.
7. On the last turn of parar o tempo repeat steps 1-4

As long as you continually roll 59-60 and never run out of nets, you can do this forever.

How many nets?

The highest carrying capacity a level 20 sorcerer can get is:

• A goliath, bugbear, orc, firbolg, centaur, or loxodon
• with 30 Strength (gained through Epic Boons)
• the High Magic Epic Boon
• having used a 9th level spell slot to cast aumentar / reduzir becoming Large.

This lets you carry 1,800 pounds which comes to 590 nets in 20 sacks.

Of course if you can just pull from a pile of nets, then the maximum number of nets is arbitrarily large.

However, if you are interested in a strategy to throw nets a little bit faster, see my previous strategy below:

The most a single character can throw is 37

O personagem

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

A preparação

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

1. Elenco pressa 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 o efeito de pressa to 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. Draw two nets as part of your movement.
4. Throw a net. ---------------------------------------------------------------------- [3]
5. Use War Priest to throw a net. ------------------------------------------------- [4]
6. Use o efeito de pressa to take the Use an Object action and draw two nets.
7. Use surto de ação.
8. Elenco parar o tempo rolling a 4 on the 1d4.

Turn One of Parar o tempo

9. Roll 81-82 on the d100 for Wild Magic Surge.
10. Throw a net. ---------------------------------------------------------------------- [5]
11. Elenco ray of sickness with the Quickened Spell Metamagic on an empty space (use Bend Luck)
12. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.
13. Throw a net. ---------------------------------------------------------------------- [6]
14. Draw two nets as part of your movement.
15. Throw a net at an empty space. ------------------------------------------------ [7]
16. Use o pressa action to throw a net at an empty space. --------------------- [8]

Turn Two of Parar o tempo

17. Roll 81-82 on the d100 for Wild Magic Surge.
18. Take the Use an Object action to draw two nets.
19. Throw a net at an empty space. ------------------------------------------------ [9]
20. Use War Priest to throw a net at an empty space. --------------------------- [10]
21. Draw two nets as part of your movement.
22. Use o pressa action to throw a net at an empty space. --------------------- [11]

Turn Three of Parar o tempo

23. Roll 81-82 on the d100 for Wild Magic Surge.
24. Throw a net. ---------------------------------------------------------------------- [12]
25. Elenco ray of sickness with the Quickened Spell Metamagic on an empty space (use Bend Luck).
26. Roll 81-82 on the d100 for Wild Magic Surge to recover Bend Luck.
27. Take the Use an Object action to draw two nets.
28. Throw a net at an empty space. ------------------------------------------------ [13]
29. Use o pressa action to throw a net at an empty space. --------------------- [14]
30. Draw two nets as part of your movement.

Turn Four of Parar o tempo

31. Repeat steps 9-11. --------------------------------------------------------------- [18]

Turn Five of First Parar o tempo

32. Roll 81-82 on the d100 for Wild Magic Surge.
33. Take the Use an Object action to draw two nets.
34. Use o pressa action to throw a net at an empty space. --------------------- [19]
35. Use War Priest to throw a net at an empty space. --------------------------- [20]
36. Draw two nets as part of your movement.
37. Repeat steps 8-26. -------------------------------------------------------------- [34]

Turn Five of Second Parar o tempo

38. Roll 81-82 on the d100 for Wild Magic Surge.
39. Take the Use an Object action to draw two nets.
40. Throw a net at an empty space. ----------------------------------------------- [35]
41. Use War Priest to throw a net at an empty space. -------------------------- [36]
42. Draw your last net as part of your movement.
43. Use pressa action to throw a net. ---------------------------------------------- [37]

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 pressa. 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 (keep in mind that a high level Wild Magic sorcerer can roll twice on the table and choose either result via Controlled Chaos | Thanks for the mention in the comments @Speedkat):

• 1 on the d20 one time
  • (1 / 20)
• 1 or 2 on the d100 one time
  • 1-(98/100)²
• 81 or 82 on the d100 nineteen times
  • (1-(98/100)²)¹⁹
• 4 on the d4 two times
  • (1 / 4)²

Multiplying these probablities comes to one chance in just over three nonillion (or 3,558,330,300,000,000,000,000,000,000,000 in numerical (rounded))

If we take the average number of times we roll 81-82 on the d100, then our probability drops considerably to one in 8,081, but we only get an average of 20.6732 nets

Or we can take the average result for all Wild Magic Surges dropping our probability to one in sixteen, but we only get an average of 20.001 nets

¹ You have to target an empty space with ray of sickness and the nets during parar o tempo 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 (ênfase meu):

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

4 nets in one round in those conditions

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

Rodada 1.

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

Rodada 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 Ação Surge to make another attack and throw your fourth and last net.

notas

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 pressa, 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 pressa spell to add another action allowing you to throw a fifth net. Then another one of your allies uses the Ataque do Comandante 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.
Classe: Sorcerer 5 (wild) / Rogue 3 (thief) / Fighter 2, Dual Wielder feat.

Preparatory Round:
Elenco 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)

(intended as a baseline value)

A Level-2 Fighter can make 3 net attacks.

Variant Human With the Dual Wielder feat.
Stats: S16; D8; C16; I8; W15; C8

Prep Round:

• Have two nets drawn
• Ready an action to throw a net.

Go-time:

1. Use readied action before your turn starts to throw 1 net
2. Attack Action 2nd net
3. Object interaction to draw two more nets
4. Action Surge a 3rd net
5. If a reaction attack becomes available, you can use it to throw a 4th net
   • Though, I don't know of any way to use your reaction to throw a net.