At least ten
This requires a Thief Rogue / Fighter affected by haste and with the Dual Wielder feat which states:
[...] You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.
The character, like any, can take the Use an Object action, using their action to draw two weapons and can use their item interaction to draw two more.
The Thief Rogue gets the Fast Hands feature which states:
Starting at 3rd level, you can use the bonus action granted by your Cunning Action to make a Dexterity (Sleight of Hand) check, use your thieves' tools to disarm a trap or open a lock, or take the Use an Object action.
Then we can multiclass into Fighter to get Action Surge which states:
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action.
This lets us take the Use an Object action yet again, giving us two more weapons to draw.
After that we would want the haste spell which states:
[The creature] gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
We would take the Use an Object action to draw another two weapons bringing our total to ten.
In summary we have:
Two weapons from our action, two from our object interaction, two from Action Surge, two from Fast Hands, and two from haste.