You can achieve 3 attacks per turn with a 2nd level Monk by using 1 ki point to trigger Flurry of Blows, making 1 attack with your action and 2 attacks with your bonus action.
You can achieve 3 attacks per turn with a 5th level Frenzy Barbarian: two attacks with Extra Attack, and one with your bonus action.
You can also achieve 3 attacks per turn with any class that gets Extra Attack if you use Dual Wielding: one weapon in each hand, attacking twice with one weapon using your action, and once with the other weapon using your bonus action.
You can also achieve 3 attacks per turn with a 6th level Bladesinger Wizard: with Haste cast on yourself as well as Extra Attack, you can attack twice with your normal action, and once with your Haste action.
You can also achieve 3 attacks per turn with a 10th level Lore Bard: Swift Quiver to attack twice with your bonus action, and once with your action.
And finally, as stated, 11 levels of Fighter grants you 3 attacks.
Multiattack is not the same as Extra Attack
Note that Master Thief has Multiattack:
The thief makes three attacks with its shortsword.
This means it cannot actually attack with any other weapon thrice. On the other hand, Extra Attack allows you to attack with any number of weapons as many number attacks as your Extra Attack feature allows.
Also, you can replace an attack from your Extra Attack with a grapple or shove check, whereas Multiattack does not allow this.
In other words, from a strict mechanical point of view, you cannot build a Player Character like a Master Thief.